Ceva
Geogebra sheet
Coq statement
Lemma Ceva: forall A B C D E F M:point,
collinear M A D -> collinear M B E -> collinear M C F ->
collinear B C D -> collinear E A C -> collinear F A B ->
(distance2 D B) * (distance2 E C) * (distance2 F A) =
(distance2 D C) * (distance2 E A) * (distance2 F B)
\/ collinear A B C.
Proof. geo_begin. tzR. Qed.
Algebraic version
p:= ((-1)) *a^2*b^2*d*e^2*f + ((-1)) *b^2*c^2*d*e^2*f + ((-1)) *a^2*d^3*e^2*f + ((-1)) *c^2*d^3*e^2*f + (2) *b^2*c*d*e^2*f^2 + (2) *c*d^3*e^2*f^2 + (1) *a^2*b^2*d*f^3 + (1) *b^2*c^2*d*f^3 + (1) *a^2*d^3*f^3 + (1) *c^2*d^3*f^3 + ((-2)) *a^2*d^2*e*f^3 + ((-2)) *c^2*d^2*e*f^3 + (1) *a^2*d*e^2*f^3 + ((-1)) *b^2*d*e^2*f^3 + (1) *c^2*d*e^2*f^3 + ((-1)) *d^3*e^2*f^3 + ((-2)) *a^2*b*d*f^3*g + ((-2)) *b*c^2*d*f^3*g + ((-1)) *a^2*b^2*d*f*g^2 + ((-1)) *b^2*c^2*d*f*g^2 + ((-1)) *a^2*d^3*f*g^2 + ((-1)) *c^2*d^3*f*g^2 + (2) *b^2*c*d*f^2*g^2 + (2) *c*d^3*f^2*g^2 + (1) *a^2*d*f^3*g^2 + ((-1)) *b^2*d*f^3*g^2 + (1) *c^2*d*f^3*g^2 + ((-1)) *d^3*f^3*g^2 + (1) *a^2*b^3*e^2*h + (1) *b^3*c^2*e^2*h + (1) *a^2*b*d^2*e^2*h + (1) *b*c^2*d^2*e^2*h + ((-2)) *b^3*c*e^2*f*h + (2) *a*b^2*d*e^2*f*h + ((-2)) *b*c*d^2*e^2*f*h + (2) *a*d^3*e^2*f*h + ((-1)) *a^2*b^3*f^2*h + ((-1)) *b^3*c^2*f^2*h + ((-1)) *a^2*b*d^2*f^2*h + ((-1)) *b*c^2*d^2*f^2*h + (2) *a^2*b*d*e*f^2*h + (2) *b*c^2*d*e*f^2*h + ((-1)) *a^2*b*e^2*f^2*h + (1) *b^3*e^2*f^2*h + ((-1)) *b*c^2*e^2*f^2*h + (1) *b*d^2*e^2*f^2*h + (2) *a^2*b^2*f^2*g*h + (2) *b^2*c^2*f^2*g*h + (1) *a^2*b^3*g^2*h + (1) *b^3*c^2*g^2*h + (1) *a^2*b*d^2*g^2*h + (1) *b*c^2*d^2*g^2*h + ((-2)) *b^3*c*f*g^2*h + (2) *a*b^2*d*f*g^2*h + ((-2)) *b*c*d^2*f*g^2*h + (2) *a*d^3*f*g^2*h + ((-1)) *a^2*b*f^2*g^2*h + (1) *b^3*f^2*g^2*h + ((-1)) *b*c^2*f^2*g^2*h + (1) *b*d^2*f^2*g^2*h + ((-2)) *a*b^3*e^2*h^2 + ((-2)) *a*b*d^2*e^2*h^2 + (1) *a^2*b^2*d*f*h^2 + (1) *b^2*c^2*d*f*h^2 + (1) *a^2*d^3*f*h^2 + (1) *c^2*d^3*f*h^2 + ((-2)) *a^2*d^2*e*f*h^2 + ((-2)) *c^2*d^2*e*f*h^2 + (1) *a^2*d*e^2*f*h^2 + ((-1)) *b^2*d*e^2*f*h^2 + (1) *c^2*d*e^2*f*h^2 + ((-1)) *d^3*e^2*f*h^2 + ((-2)) *a^2*b*d*f*g*h^2 + ((-2)) *b*c^2*d*f*g*h^2 + ((-2)) *a*b^3*g^2*h^2 + ((-2)) *a*b*d^2*g^2*h^2 + (1) *a^2*d*f*g^2*h^2 + ((-1)) *b^2*d*f*g^2*h^2 + (1) *c^2*d*f*g^2*h^2 + ((-1)) *d^3*f*g^2*h^2 + ((-1)) *a^2*b^3*h^3 + ((-1)) *b^3*c^2*h^3 + ((-1)) *a^2*b*d^2*h^3 + ((-1)) *b*c^2*d^2*h^3 + (2) *a^2*b*d*e*h^3 + (2) *b*c^2*d*e*h^3 + ((-1)) *a^2*b*e^2*h^3 + (1) *b^3*e^2*h^3 + ((-1)) *b*c^2*e^2*h^3 + (1) *b*d^2*e^2*h^3 + (2) *a^2*b^2*g*h^3 + (2) *b^2*c^2*g*h^3 + ((-1)) *a^2*b*g^2*h^3 + (1) *b^3*g^2*h^3 + ((-1)) *b*c^2*g^2*h^3 + (1) *b*d^2*g^2*h^3 + (2) *a^2*d^2*e^2*f*i + (2) *c^2*d^2*e^2*f*i + ((-4)) *c*d^2*e^2*f^2*i + (2) *d^2*e^2*f^3*i + (2) *a^2*d^2*f*g^2*i + (2) *c^2*d^2*f*g^2*i + ((-4)) *c*d^2*f^2*g^2*i + (2) *d^2*f^3*g^2*i + ((-2)) *a^2*b*d*e^2*h*i + ((-2)) *b*c^2*d*e^2*h*i + ((-2)) *a^2*b^2*d*f*h*i + ((-2)) *b^2*c^2*d*f*h*i + ((-2)) *a^2*d^3*f*h*i + ((-2)) *c^2*d^3*f*h*i + (4) *a^2*d^2*e*f*h*i + (4) *c^2*d^2*e*f*h*i + ((-2)) *a^2*d*e^2*f*h*i + (4) *b*c*d*e^2*f*h*i + ((-2)) *c^2*d*e^2*f*h*i + ((-4)) *a*d^2*e^2*f*h*i + ((-2)) *b*d*e^2*f^2*h*i + (4) *a^2*b*d*f*g*h*i + (4) *b*c^2*d*f*g*h*i + ((-2)) *a^2*b*d*g^2*h*i + ((-2)) *b*c^2*d*g^2*h*i + ((-2)) *a^2*d*f*g^2*h*i + (4) *b*c*d*f*g^2*h*i + ((-2)) *c^2*d*f*g^2*h*i + ((-4)) *a*d^2*f*g^2*h*i + ((-2)) *b*d*f^2*g^2*h*i + (2) *a^2*b^3*h^2*i + (2) *b^3*c^2*h^2*i + (2) *a^2*b*d^2*h^2*i + (2) *b*c^2*d^2*h^2*i + ((-4)) *a^2*b*d*e*h^2*i + ((-4)) *b*c^2*d*e*h^2*i + (2) *a^2*b*e^2*h^2*i + (2) *b*c^2*e^2*h^2*i + (4) *a*b*d*e^2*h^2*i + (2) *d^2*e^2*f*h^2*i + ((-4)) *a^2*b^2*g*h^2*i + ((-4)) *b^2*c^2*g*h^2*i + (2) *a^2*b*g^2*h^2*i + (2) *b*c^2*g^2*h^2*i + (4) *a*b*d*g^2*h^2*i + (2) *d^2*f*g^2*h^2*i + ((-2)) *b*d*e^2*h^3*i + ((-2)) *b*d*g^2*h^3*i + (1) *a^2*b^2*d*f*i^2 + (1) *b^2*c^2*d*f*i^2 + (1) *a^2*d^3*f*i^2 + (1) *c^2*d^3*f*i^2 + ((-2)) *a^2*d^2*e*f*i^2 + ((-2)) *c^2*d^2*e*f*i^2 + (2) *c*d*e^2*f^2*i^2 + ((-1)) *d*e^2*f^3*i^2 + ((-2)) *a^2*b*d*f*g*i^2 + ((-2)) *b*c^2*d*f*g*i^2 + (2) *c*d*f^2*g^2*i^2 + ((-1)) *d*f^3*g^2*i^2 + ((-1)) *a^2*b^3*h*i^2 + ((-1)) *b^3*c^2*h*i^2 + ((-1)) *a^2*b*d^2*h*i^2 + ((-1)) *b*c^2*d^2*h*i^2 + (2) *a^2*b*d*e*h*i^2 + (2) *b*c^2*d*e*h*i^2 + ((-2)) *b*c*e^2*f*h*i^2 + (2) *a*d*e^2*f*h*i^2 + (1) *b*e^2*f^2*h*i^2 + (2) *a^2*b^2*g*h*i^2 + (2) *b^2*c^2*g*h*i^2 + ((-2)) *b*c*f*g^2*h*i^2 + (2) *a*d*f*g^2*h*i^2 + (1) *b*f^2*g^2*h*i^2 + ((-2)) *a*b*e^2*h^2*i^2 + ((-1)) *d*e^2*f*h^2*i^2 + ((-2)) *a*b*g^2*h^2*i^2 + ((-1)) *d*f*g^2*h^2*i^2 + (1) *b*e^2*h^3*i^2 + (1) *b*g^2*h^3*i^2 + (1) *a^2*b^2*d*e^2*k + (1) *b^2*c^2*d*e^2*k + (1) *a^2*d^3*e^2*k + (1) *c^2*d^3*e^2*k + ((-2)) *b^2*c*d*e^2*f*k + ((-2)) *c*d^3*e^2*f*k + ((-1)) *a^2*b^2*d*f^2*k + ((-1)) *b^2*c^2*d*f^2*k + ((-1)) *a^2*d^3*f^2*k + ((-1)) *c^2*d^3*f^2*k + (2) *a^2*d^2*e*f^2*k + (2) *c^2*d^2*e*f^2*k + ((-1)) *a^2*d*e^2*f^2*k + (1) *b^2*d*e^2*f^2*k + ((-1)) *c^2*d*e^2*f^2*k + (1) *d^3*e^2*f^2*k + ((-2)) *b^2*c*d*f^3*k + ((-2)) *c*d^3*f^3*k + (4) *c*d^2*e*f^3*k + ((-2)) *c*d*e^2*f^3*k + (2) *a^2*b^2*d*f*g*k + (2) *b^2*c^2*d*f*g*k + (2) *a^2*d^3*f*g*k + (2) *c^2*d^3*f*g*k + (2) *a^2*b*d*f^2*g*k + ((-4)) *b^2*c*d*f^2*g*k + (2) *b*c^2*d*f^2*g*k + ((-4)) *c*d^3*f^2*g*k + (2) *b^2*d*f^3*g*k + (4) *b*c*d*f^3*g*k + (2) *d^3*f^3*g*k + (1) *a^2*b^2*d*g^2*k + (1) *b^2*c^2*d*g^2*k + (1) *a^2*d^3*g^2*k + (1) *c^2*d^3*g^2*k + ((-2)) *b^2*c*d*f*g^2*k + ((-2)) *c*d^3*f*g^2*k + ((-1)) *a^2*d*f^2*g^2*k + (1) *b^2*d*f^2*g^2*k + ((-1)) *c^2*d*f^2*g^2*k + (1) *d^3*f^2*g^2*k + ((-2)) *c*d*f^3*g^2*k + ((-1)) *a^2*b^2*e^2*h*k + ((-1)) *b^2*c^2*e^2*h*k + ((-2)) *a*b^2*d*e^2*h*k + ((-1)) *a^2*d^2*e^2*h*k + ((-1)) *c^2*d^2*e^2*h*k + ((-2)) *a*d^3*e^2*h*k + (2) *b^2*c*e^2*f*h*k + (2) *c*d^2*e^2*f*h*k + (1) *a^2*b^2*f^2*h*k + (2) *b^3*c*f^2*h*k + (1) *b^2*c^2*f^2*h*k + (1) *a^2*d^2*f^2*h*k + (2) *b*c*d^2*f^2*h*k + (1) *c^2*d^2*f^2*h*k + ((-2)) *a^2*d*e*f^2*h*k + ((-4)) *b*c*d*e*f^2*h*k + ((-2)) *c^2*d*e*f^2*h*k + (1) *a^2*e^2*f^2*h*k + ((-1)) *b^2*e^2*f^2*h*k + (2) *b*c*e^2*f^2*h*k + (1) *c^2*e^2*f^2*h*k + ((-1)) *d^2*e^2*f^2*h*k + ((-2)) *a^2*b^3*g*h*k + ((-2)) *b^3*c^2*g*h*k + ((-2)) *a^2*b*d^2*g*h*k + ((-2)) *b*c^2*d^2*g*h*k + (4) *b^3*c*f*g*h*k + ((-4)) *a*b^2*d*f*g*h*k + (4) *b*c*d^2*f*g*h*k + ((-4)) *a*d^3*f*g*h*k + ((-2)) *a^2*b*f^2*g*h*k + ((-2)) *b^3*f^2*g*h*k + ((-4)) *b^2*c*f^2*g*h*k + ((-2)) *b*c^2*f^2*g*h*k + ((-2)) *b*d^2*f^2*g*h*k + ((-1)) *a^2*b^2*g^2*h*k + ((-1)) *b^2*c^2*g^2*h*k + ((-2)) *a*b^2*d*g^2*h*k + ((-1)) *a^2*d^2*g^2*h*k + ((-1)) *c^2*d^2*g^2*h*k + ((-2)) *a*d^3*g^2*h*k + (2) *b^2*c*f*g^2*h*k + (2) *c*d^2*f*g^2*h*k + (1) *a^2*f^2*g^2*h*k + ((-1)) *b^2*f^2*g^2*h*k + (2) *b*c*f^2*g^2*h*k + (1) *c^2*f^2*g^2*h*k + ((-1)) *d^2*f^2*g^2*h*k + ((-1)) *a^2*b^2*d*h^2*k + ((-1)) *b^2*c^2*d*h^2*k + ((-1)) *a^2*d^3*h^2*k + ((-1)) *c^2*d^3*h^2*k + (2) *a^2*d^2*e*h^2*k + (2) *c^2*d^2*e*h^2*k + (2) *a*b^2*e^2*h^2*k + ((-1)) *a^2*d*e^2*h^2*k + (1) *b^2*d*e^2*h^2*k + ((-1)) *c^2*d*e^2*h^2*k + (2) *a*d^2*e^2*h^2*k + (1) *d^3*e^2*h^2*k + ((-2)) *b^2*c*d*f*h^2*k + ((-2)) *c*d^3*f*h^2*k + (4) *c*d^2*e*f*h^2*k + ((-2)) *c*d*e^2*f*h^2*k + (4) *a*b^3*g*h^2*k + (2) *a^2*b*d*g*h^2*k + (2) *b*c^2*d*g*h^2*k + (4) *a*b*d^2*g*h^2*k + (2) *b^2*d*f*g*h^2*k + (4) *b*c*d*f*g*h^2*k + (2) *d^3*f*g*h^2*k + (2) *a*b^2*g^2*h^2*k + ((-1)) *a^2*d*g^2*h^2*k + (1) *b^2*d*g^2*h^2*k + ((-1)) *c^2*d*g^2*h^2*k + (2) *a*d^2*g^2*h^2*k + (1) *d^3*g^2*h^2*k + ((-2)) *c*d*f*g^2*h^2*k + (1) *a^2*b^2*h^3*k + (2) *b^3*c*h^3*k + (1) *b^2*c^2*h^3*k + (1) *a^2*d^2*h^3*k + (2) *b*c*d^2*h^3*k + (1) *c^2*d^2*h^3*k + ((-2)) *a^2*d*e*h^3*k + ((-4)) *b*c*d*e*h^3*k + ((-2)) *c^2*d*e*h^3*k + (1) *a^2*e^2*h^3*k + ((-1)) *b^2*e^2*h^3*k + (2) *b*c*e^2*h^3*k + (1) *c^2*e^2*h^3*k + ((-1)) *d^2*e^2*h^3*k + ((-2)) *a^2*b*g*h^3*k + ((-2)) *b^3*g*h^3*k + ((-4)) *b^2*c*g*h^3*k + ((-2)) *b*c^2*g*h^3*k + ((-2)) *b*d^2*g*h^3*k + (1) *a^2*g^2*h^3*k + ((-1)) *b^2*g^2*h^3*k + (2) *b*c*g^2*h^3*k + (1) *c^2*g^2*h^3*k + ((-1)) *d^2*g^2*h^3*k + ((-2)) *a^2*d^2*e^2*i*k + ((-2)) *c^2*d^2*e^2*i*k + (4) *c*d^2*e^2*f*i*k + ((-2)) *d^2*e^2*f^2*i*k + ((-4)) *a^2*d^2*f*g*i*k + ((-4)) *c^2*d^2*f*g*i*k + (8) *c*d^2*f^2*g*i*k + ((-4)) *d^2*f^3*g*i*k + ((-2)) *a^2*d^2*g^2*i*k + ((-2)) *c^2*d^2*g^2*i*k + (4) *c*d^2*f*g^2*i*k + ((-2)) *d^2*f^2*g^2*i*k + (2) *a^2*b^2*d*h*i*k + (2) *b^2*c^2*d*h*i*k + (2) *a^2*d^3*h*i*k + (2) *c^2*d^3*h*i*k + ((-4)) *a^2*d^2*e*h*i*k + ((-4)) *c^2*d^2*e*h*i*k + (4) *a^2*d*e^2*h*i*k + (4) *c^2*d*e^2*h*i*k + (4) *a*d^2*e^2*h*i*k + (4) *b^2*c*d*f*h*i*k + (4) *c*d^3*f*h*i*k + ((-8)) *c*d^2*e*f*h*i*k + (2) *d*e^2*f^2*h*i*k + ((-16)) *b*c*d*f*g*h*i*k + (8) *a*d^2*f*g*h*i*k + (4) *b*d*f^2*g*h*i*k + (4) *a^2*d*g^2*h*i*k + (4) *c^2*d*g^2*h*i*k + (4) *a*d^2*g^2*h*i*k + (2) *d*f^2*g^2*h*i*k + ((-2)) *a^2*b^2*h^2*i*k + ((-4)) *b^3*c*h^2*i*k + ((-2)) *b^2*c^2*h^2*i*k + ((-2)) *a^2*d^2*h^2*i*k + ((-4)) *b*c*d^2*h^2*i*k + ((-2)) *c^2*d^2*h^2*i*k + (4) *a^2*d*e*h^2*i*k + (8) *b*c*d*e*h^2*i*k + (4) *c^2*d*e*h^2*i*k + ((-2)) *a^2*e^2*h^2*i*k + ((-4)) *b*c*e^2*h^2*i*k + ((-2)) *c^2*e^2*h^2*i*k + ((-4)) *a*d*e^2*h^2*i*k + ((-2)) *d^2*e^2*h^2*i*k + (4) *a^2*b*g*h^2*i*k + (8) *b^2*c*g*h^2*i*k + (4) *b*c^2*g*h^2*i*k + ((-8)) *a*b*d*g*h^2*i*k + ((-4)) *d^2*f*g*h^2*i*k + ((-2)) *a^2*g^2*h^2*i*k + ((-4)) *b*c*g^2*h^2*i*k + ((-2)) *c^2*g^2*h^2*i*k + ((-4)) *a*d*g^2*h^2*i*k + ((-2)) *d^2*g^2*h^2*i*k + (2) *d*e^2*h^3*i*k + (4) *b*d*g*h^3*i*k + (2) 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*a*d^3*h*k*m^2 + (2) *b^2*c*f*h*k*m^2 + (2) *c*d^2*f*h*k*m^2 + ((-2)) *d*e*f^2*h*k*m^2 + (1) *e^2*f^2*h*k*m^2 + ((-2)) *b*f^2*g*h*k*m^2 + (1) *f^2*g^2*h*k*m^2 + (2) *a*b^2*h^2*k*m^2 + (2) *a*d^2*h^2*k*m^2 + (2) *d^2*e*h^2*k*m^2 + ((-1)) *d*e^2*h^2*k*m^2 + (2) *b*d*g*h^2*k*m^2 + ((-1)) *d*g^2*h^2*k*m^2 + ((-2)) *d*e*h^3*k*m^2 + (1) *e^2*h^3*k*m^2 + ((-2)) *b*g*h^3*k*m^2 + (1) *g^2*h^3*k*m^2 + ((-2)) *a^2*d^2*i*k*m^2 + ((-2)) *c^2*d^2*i*k*m^2 + (4) *c*d^2*f*i*k*m^2 + ((-2)) *d^2*f^2*i*k*m^2 + (2) *a^2*d*h*i*k*m^2 + (2) *b^2*d*h*i*k*m^2 + (2) *c^2*d*h*i*k*m^2 + (4) *a*d^2*h*i*k*m^2 + (2) *d^3*h*i*k*m^2 + ((-4)) *d^2*e*h*i*k*m^2 + (2) *d*e^2*h*i*k*m^2 + ((-4)) *c*d*f*h*i*k*m^2 + (2) *d*f^2*h*i*k*m^2 + ((-4)) *b*d*g*h*i*k*m^2 + (2) *d*g^2*h*i*k*m^2 + ((-2)) *b^2*h^2*i*k*m^2 + ((-4)) *a*d*h^2*i*k*m^2 + ((-4)) *d^2*h^2*i*k*m^2 + (4) *d*e*h^2*i*k*m^2 + ((-2)) *e^2*h^2*i*k*m^2 + (4) *b*g*h^2*i*k*m^2 + ((-2)) *g^2*h^2*i*k*m^2 + (2) *d*h^3*i*k*m^2 + (1) *a^2*d*i^2*k*m^2 + ((-1)) *b^2*d*i^2*k*m^2 + (1) *c^2*d*i^2*k*m^2 + ((-1)) *d^3*i^2*k*m^2 + (2) *d^2*e*i^2*k*m^2 + ((-1)) *d*e^2*i^2*k*m^2 + ((-2)) *c*d*f*i^2*k*m^2 + (1) *d*f^2*i^2*k*m^2 + (2) *b*d*g*i^2*k*m^2 + ((-1)) *d*g^2*i^2*k*m^2 + ((-1)) *a^2*h*i^2*k*m^2 + (1) *b^2*h*i^2*k*m^2 + ((-1)) *c^2*h*i^2*k*m^2 + ((-2)) *a*d*h*i^2*k*m^2 + (1) *d^2*h*i^2*k*m^2 + ((-2)) *d*e*h*i^2*k*m^2 + (1) *e^2*h*i^2*k*m^2 + (2) *c*f*h*i^2*k*m^2 + ((-1)) *f^2*h*i^2*k*m^2 + ((-2)) *b*g*h*i^2*k*m^2 + (1) *g^2*h*i^2*k*m^2 + (2) *a*h^2*i^2*k*m^2 + (1) *d*h^2*i^2*k*m^2 + ((-1)) *h^3*i^2*k*m^2 + ((-4)) *a^2*b^2*e*l*m^2 + ((-4)) *b^2*c^2*e*l*m^2 + (4) *a*b^3*f*l*m^2 + (2) *a^2*b*d*f*l*m^2 + (2) *b*c^2*d*f*l*m^2 + (4) *a*b*d^2*f*l*m^2 + (4) *a^2*b*e*f*l*m^2 + (8) *b^2*c*e*f*l*m^2 + (4) *b*c^2*e*f*l*m^2 + ((-8)) *a*b*d*e*f*l*m^2 + (4) *a*b*e^2*f*l*m^2 + ((-4)) *a*b^2*f^2*l*m^2 + ((-2)) *b^2*d*f^2*l*m^2 + ((-4)) *b*c*d*f^2*l*m^2 + ((-4)) *a*d^2*f^2*l*m^2 + ((-2)) *d^3*f^2*l*m^2 + ((-4)) *b^2*e*f^2*l*m^2 + ((-8)) *b*c*e*f^2*l*m^2 + (8) *a*d*e*f^2*l*m^2 + (4) *d^2*e*f^2*l*m^2 + ((-4)) *a*e^2*f^2*l*m^2 + ((-2)) *d*e^2*f^2*l*m^2 + (2) *b*d*f^3*l*m^2 + (4) *b*e*f^3*l*m^2 + ((-8)) *a*b^2*f*g*l*m^2 + (8) *a*b*f^2*g*l*m^2 + (4) *b*d*f^2*g*l*m^2 + (4) *a*b*f*g^2*l*m^2 + ((-4)) *a*f^2*g^2*l*m^2 + ((-2)) *d*f^2*g^2*l*m^2 + ((-2)) *a^2*b^2*h*l*m^2 + ((-2)) *b^2*c^2*h*l*m^2 + (8) *a*b^2*e*h*l*m^2 + (2) *b^3*f*h*l*m^2 + (4) *b^2*c*f*h*l*m^2 + ((-4)) *a*b*d*f*h*l*m^2 + (2) *b*d^2*f*h*l*m^2 + ((-8)) *a*b*e*f*h*l*m^2 + ((-4)) *b*d*e*f*h*l*m^2 + (2) *b*e^2*f*h*l*m^2 + ((-2)) *b^2*f^2*h*l*m^2 + ((-4)) *b^2*f*g*h*l*m^2 + (2) *b*f*g^2*h*l*m^2 + (4) *a*b^2*h^2*l*m^2 + ((-4)) *b^2*e*h^2*l*m^2 + (2) *b*d*f*h^2*l*m^2 + (4) *b*e*f*h^2*l*m^2 + ((-2)) *b^2*h^3*l*m^2 + ((-2)) *a^2*b*d*k*l*m^2 + ((-2)) *b*c^2*d*k*l*m^2 + (2) *b^2*d*f*k*l*m^2 + (4) *b*c*d*f*k*l*m^2 + (2) *d^3*f*k*l*m^2 + ((-4)) *d^2*e*f*k*l*m^2 + (2) *d*e^2*f*k*l*m^2 + ((-2)) *b*d*f^2*k*l*m^2 + ((-4)) *b*d*f*g*k*l*m^2 + (2) *d*f*g^2*k*l*m^2 + (2) *a^2*b*h*k*l*m^2 + (2) *b*c^2*h*k*l*m^2 + (4) *a*b*d*h*k*l*m^2 + ((-2)) *b^2*f*h*k*l*m^2 + ((-4)) *b*c*f*h*k*l*m^2 + ((-2)) *d^2*f*h*k*l*m^2 + (4) *d*e*f*h*k*l*m^2 + ((-2)) *e^2*f*h*k*l*m^2 + (2) *b*f^2*h*k*l*m^2 + (4) *b*f*g*h*k*l*m^2 + ((-2)) *f*g^2*h*k*l*m^2 + ((-4)) *a*b*h^2*k*l*m^2 + ((-2)) *b*d*h^2*k*l*m^2 + (2) *b*h^3*k*l*m^2 + ((-2)) *a*b^3*l^2*m^2 + ((-2)) *a*b*d^2*l^2*m^2 + (2) *a^2*b*e*l^2*m^2 + (2) *b*c^2*e*l^2*m^2 + (4) *a*b*d*e*l^2*m^2 + ((-2)) *a*b*e^2*l^2*m^2 + (2) *a*b^2*f*l^2*m^2 + ((-1)) *a^2*d*f*l^2*m^2 + (1) *b^2*d*f*l^2*m^2 + ((-1)) *c^2*d*f*l^2*m^2 + (2) *a*d^2*f*l^2*m^2 + (1) *d^3*f*l^2*m^2 + ((-2)) *a^2*e*f*l^2*m^2 + ((-4)) *b*c*e*f*l^2*m^2 + ((-2)) *c^2*e*f*l^2*m^2 + ((-4)) *a*d*e*f*l^2*m^2 + ((-2)) *d^2*e*f*l^2*m^2 + (2) *a*e^2*f*l^2*m^2 + (1) *d*e^2*f*l^2*m^2 + (2) *c*d*f^2*l^2*m^2 + (2) *b*e*f^2*l^2*m^2 + (4) *c*e*f^2*l^2*m^2 + ((-1)) *d*f^3*l^2*m^2 + ((-2)) *e*f^3*l^2*m^2 + (4) *a*b^2*g*l^2*m^2 + ((-4)) *a*b*f*g*l^2*m^2 + ((-2)) *b*d*f*g*l^2*m^2 + ((-2)) *a*b*g^2*l^2*m^2 + (2) *a*f*g^2*l^2*m^2 + (1) *d*f*g^2*l^2*m^2 + (1) *a^2*b*h*l^2*m^2 + ((-1)) *b^3*h*l^2*m^2 + (1) *b*c^2*h*l^2*m^2 + ((-1)) *b*d^2*h*l^2*m^2 + ((-4)) *a*b*e*h*l^2*m^2 + (2) *b*d*e*h*l^2*m^2 + ((-1)) *b*e^2*h*l^2*m^2 + ((-2)) *b*c*f*h*l^2*m^2 + (2) *a*d*f*h*l^2*m^2 + (4) *a*e*f*h*l^2*m^2 + (1) *b*f^2*h*l^2*m^2 + (2) *b^2*g*h*l^2*m^2 + ((-1)) *b*g^2*h*l^2*m^2 + ((-2)) *a*b*h^2*l^2*m^2 + (2) *b*e*h^2*l^2*m^2 + ((-1)) *d*f*h^2*l^2*m^2 + ((-2)) *e*f*h^2*l^2*m^2 + (1) *b*h^3*l^2*m^2 + (1) *a^2*d*k*l^2*m^2 + ((-1)) *b^2*d*k*l^2*m^2 + (1) *c^2*d*k*l^2*m^2 + ((-1)) *d^3*k*l^2*m^2 + (2) *d^2*e*k*l^2*m^2 + ((-1)) *d*e^2*k*l^2*m^2 + ((-2)) *c*d*f*k*l^2*m^2 + (1) *d*f^2*k*l^2*m^2 + (2) *b*d*g*k*l^2*m^2 + ((-1)) *d*g^2*k*l^2*m^2 + ((-1)) *a^2*h*k*l^2*m^2 + (1) *b^2*h*k*l^2*m^2 + ((-1)) *c^2*h*k*l^2*m^2 + ((-2)) *a*d*h*k*l^2*m^2 + (1) *d^2*h*k*l^2*m^2 + ((-2)) *d*e*h*k*l^2*m^2 + (1) *e^2*h*k*l^2*m^2 + (2) *c*f*h*k*l^2*m^2 + ((-1)) *f^2*h*k*l^2*m^2 + ((-2)) *b*g*h*k*l^2*m^2 + (1) *g^2*h*k*l^2*m^2 + (2) *a*h^2*k*l^2*m^2 + (1) *d*h^2*k*l^2*m^2 + ((-1)) *h^3*k*l^2*m^2 + ((-1)) *a^2*b^3*m^3 + ((-1)) *b^3*c^2*m^3 + ((-1)) *a^2*b*d^2*m^3 + ((-1)) *b*c^2*d^2*m^3 + (1) *a^2*b^2*f*m^3 + (2) *b^3*c*f*m^3 + (1) *b^2*c^2*f*m^3 + (1) *a^2*d^2*f*m^3 + (2) *b*c*d^2*f*m^3 + (1) *c^2*d^2*f*m^3 + ((-2)) *b^2*c*f^2*m^3 + ((-2)) *c*d^2*f^2*m^3 + ((-2)) *b*d*e*f^2*m^3 + (1) *b*e^2*f^2*m^3 + (2) *d*e*f^3*m^3 + ((-1)) *e^2*f^3*m^3 + ((-2)) *b^2*f^2*g*m^3 + (2) *b*f^3*g*m^3 + (1) *b*f^2*g^2*m^3 + ((-1)) *f^3*g^2*m^3 + (2) *a*b^3*h*m^3 + (2) *a*b*d^2*h*m^3 + ((-2)) *a*b^2*f*h*m^3 + ((-2)) *a*d^2*f*h*m^3 + ((-2)) *b*d*e*h^2*m^3 + (1) *b*e^2*h^2*m^3 + (2) *d*e*f*h^2*m^3 + ((-1)) *e^2*f*h^2*m^3 + ((-2)) *b^2*g*h^2*m^3 + (2) *b*f*g*h^2*m^3 + (1) *b*g^2*h^2*m^3 + ((-1)) *f*g^2*h^2*m^3 + (2) *a^2*b*d*i*m^3 + (2) *b*c^2*d*i*m^3 + ((-2)) *a^2*d*f*i*m^3 + ((-4)) *b*c*d*f*i*m^3 + ((-2)) *c^2*d*f*i*m^3 + (2) *b*d*f^2*i*m^3 + (4) *c*d*f^2*i*m^3 + ((-2)) *d*f^3*i*m^3 + ((-2)) *b^3*h*i*m^3 + ((-4)) *a*b*d*h*i*m^3 + ((-2)) *b*d^2*h*i*m^3 + (4) *b*d*e*h*i*m^3 + ((-2)) *b*e^2*h*i*m^3 + (2) *b^2*f*h*i*m^3 + (4) *a*d*f*h*i*m^3 + (2) *d^2*f*h*i*m^3 + ((-4)) *d*e*f*h*i*m^3 + (2) *e^2*f*h*i*m^3 + (4) *b^2*g*h*i*m^3 + ((-4)) *b*f*g*h*i*m^3 + ((-2)) *b*g^2*h*i*m^3 + (2) *f*g^2*h*i*m^3 + (2) *b*d*h^2*i*m^3 + ((-2)) *d*f*h^2*i*m^3 + ((-1)) *a^2*b*i^2*m^3 + (1) *b^3*i^2*m^3 + ((-1)) *b*c^2*i^2*m^3 + (1) *b*d^2*i^2*m^3 + ((-2)) *b*d*e*i^2*m^3 + (1) *b*e^2*i^2*m^3 + (1) *a^2*f*i^2*m^3 + ((-1)) *b^2*f*i^2*m^3 + (2) *b*c*f*i^2*m^3 + (1) *c^2*f*i^2*m^3 + ((-1)) *d^2*f*i^2*m^3 + (2) *d*e*f*i^2*m^3 + ((-1)) *e^2*f*i^2*m^3 + ((-1)) *b*f^2*i^2*m^3 + ((-2)) *c*f^2*i^2*m^3 + (1) *f^3*i^2*m^3 + ((-2)) *b^2*g*i^2*m^3 + (2) *b*f*g*i^2*m^3 + (1) *b*g^2*i^2*m^3 + ((-1)) *f*g^2*i^2*m^3 + (2) *a*b*h*i^2*m^3 + ((-2)) *a*f*h*i^2*m^3 + ((-1)) *b*h^2*i^2*m^3 + (1) *f*h^2*i^2*m^3 + (2) *a^2*b^2*l*m^3 + (2) *b^2*c^2*l*m^3 + ((-2)) *a^2*b*f*l*m^3 + ((-2)) *b^3*f*l*m^3 + ((-4)) *b^2*c*f*l*m^3 + ((-2)) *b*c^2*f*l*m^3 + ((-2)) *b*d^2*f*l*m^3 + (4) *b*d*e*f*l*m^3 + ((-2)) *b*e^2*f*l*m^3 + (4) *b^2*f^2*l*m^3 + (4) *b*c*f^2*l*m^3 + (2) *d^2*f^2*l*m^3 + ((-4)) *d*e*f^2*l*m^3 + (2) *e^2*f^2*l*m^3 + ((-2)) *b*f^3*l*m^3 + (4) *b^2*f*g*l*m^3 + ((-4)) *b*f^2*g*l*m^3 + ((-2)) *b*f*g^2*l*m^3 + (2) *f^2*g^2*l*m^3 + ((-4)) *a*b^2*h*l*m^3 + (4) *a*b*f*h*l*m^3 + (2) *b^2*h^2*l*m^3 + ((-2)) *b*f*h^2*l*m^3 + ((-1)) *a^2*b*l^2*m^3 + (1) *b^3*l^2*m^3 + ((-1)) *b*c^2*l^2*m^3 + (1) *b*d^2*l^2*m^3 + ((-2)) *b*d*e*l^2*m^3 + (1) *b*e^2*l^2*m^3 + (1) *a^2*f*l^2*m^3 + ((-1)) *b^2*f*l^2*m^3 + (2) *b*c*f*l^2*m^3 + (1) *c^2*f*l^2*m^3 + ((-1)) *d^2*f*l^2*m^3 + (2) *d*e*f*l^2*m^3 + ((-1)) *e^2*f*l^2*m^3 + ((-1)) *b*f^2*l^2*m^3 + ((-2)) *c*f^2*l^2*m^3 + (1) *f^3*l^2*m^3 + ((-2)) *b^2*g*l^2*m^3 + (2) *b*f*g*l^2*m^3 + (1) *b*g^2*l^2*m^3 + ((-1)) *f*g^2*l^2*m^3 + (2) *a*b*h*l^2*m^3 + ((-2)) *a*f*h*l^2*m^3 + ((-1)) *b*h^2*l^2*m^3 + (1) *f*h^2*l^2*m^3
F:= [
((-1)) *i*j + (1) *i*k + (1) *j*m + ((-1)) *l*m + ((-1)) *k*n + (1) *l*n ;
(1) *e*f + ((-1)) *g*h + ((-1)) *e*j + (1) *h*j + ((-1)) *f*n + (1) *g*n ;
(1) *a*b + ((-1)) *c*d + ((-1)) *a*j + (1) *d*j + ((-1)) *b*n + (1) *c*n ;
(1) *d*f + ((-1)) *b*h + (1) *b*i + ((-1)) *f*i + ((-1)) *d*l + (1) *h*l ;
(1) *b*e + ((-1)) *d*g + (1) *d*k + ((-1)) *e*k + ((-1)) *b*m + (1) *g*m ;
(1) *a*f + ((-1)) *c*h + ((-1)) *a*k + (1) *h*k + (1) *c*m + ((-1)) *f*m ;
];
Certificate
C:=[
[
(1) *g*i + (5) *i*k + (11) *k*m + ((-4)) *l*m + ((-2)) *g*n + ((-23)) *k*n + (6) *l*n ; ((-1)) *a*k + (1) *a*l + (2) *k*m + ((-1)) *l*m ; ((-2)) *c*g*h + ((-2)) *a*h^2 + (2) *e*h^2 + (2) *h^3 + (2) *c*g*i + (1) *a*h*i + ((-1)) *e*h*i + ((-1)) *h^2*i + (1) *a*g*k + (15) *c*h*k + (7) *g*h*k + ((-8)) *c*i*k + ((-2)) *g*i*k + ((-13)) *a*k^2 + ((-28)) *h*k^2 + (2) *a*g*l + ((-2)) *c*h*l + ((-2)) *g*h*l + ((-2)) *c*i*l + (10) *a*k*l + (2) *e*k*l + (14) *h*k*l + (10) *i*k*l + ((-1)) *a*e*m + (2) *c*g*m + ((-1)) *h^2*m + (1) *e*i*m + ((-7)) *c*k*m + (7) *g*k*m + ((-14)) *k^2*m + (9) *c*l*m + ((-11)) *g*l*m + (9) *k*l*m + ((-10)) *l^2*m + (4) *a*m^2 + (1) *e*m^2 + ((-5)) *h*m^2 + ((-2)) *i*m^2 + ((-9)) *m^3 + (1) *a*e*n + ((-2)) *c*g*n + (2) *a*h*n + ((-2)) *e*h*n + ((-4)) *h^2*n + ((-1)) *a*i*n + (2) *h*i*n + ((-7)) *c*k*n + ((-16)) *g*k*n + (65) *k^2*n + (11) *g*l*n + ((-63)) *k*l*n + (11) *l^2*n + ((-4)) *a*m*n + ((-2)) *e*m*n + (3) *h*m*n + (3) *i*m*n + (17) *m^2*n + (1) *e*n^2 + (2) *h*n^2 + ((-1)) *i*n^2 + ((-8)) *m*n^2 ; ((-2)) *c*i^2 + ((-4)) *c*g*k + (10) *i^2*k + ((-4)) *k^3 + (2) *c*g*l + (2) *c*k*l + (2) *g*k*l + (2) *k^2*l + ((-2)) *k*l^2 + ((-9)) *c*i*m + ((-1)) *g*i*m + ((-11)) *i*k*m + ((-10)) *i*l*m + ((-18)) *c*m^2 + ((-3)) *g*m^2 + (17) *k*m^2 + (12) *l*m^2 + (5) *c*i*n + ((-31)) *i*k*n + (11) *i*l*n + (24) *c*m*n + (2) *g*m*n + (3) *k*m*n + (17) *l*m*n + ((-4)) *c*n^2 + (1) *g*n^2 + (63) *k*n^2 + ((-30)) *l*n^2 ; (1) *c*f*i + ((-1)) *f^2*i + (1) *c*g*i + (1) *f*g*i + ((-2)) *c*i*k + ((-2)) *g*i*k + (2) *i*k^2 + ((-3)) *c*f*m + (1) *f^2*m + (3) *c*g*m + (1) *f*g*m + ((-4)) *c*k*m + (4) *f*k*m + ((-6)) *g*k*m + (7) *c*l*m + ((-1)) *f*l*m + (2) *g*l*m + ((-2)) *k*l*m + ((-2)) *l^2*m + ((-1)) *i*m^2 + (2) *m^3 + ((-3)) *c*g*n + ((-1)) *f*g*n + (8) *c*k*n + (2) *f*k*n + (4) *g*k*n + ((-10)) *k^2*n + ((-5)) *c*l*n + ((-1)) *f*l*n + (6) *k*l*n + ((-2)) *i*m*n + ((-8)) *m^2*n + (1) *i*n^2 + (7) *m*n^2 + ((-1)) *n^3 ; (1) *b^2*f + (1) *b*c*f + ((-1)) *b^2*g + 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[
0; 0; (1) *a ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; (1) *c*i^2*m ; 0; 0; (1) *g*i^2*m ; 0; 0; ];
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0; (1) *e ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; (1) *g*i^2*m ; 0; 0; (1) *c*i^2*m ; 0; ];
[
0; 0; 0; 0; 0; 0; 0; 0; 0; 0; (1) *i ; 0; 0; ((-1)) *h*m ; 0; ((-1)) *i*m ; 0; 0; ((-1)) *a*m ; ((-1)) *h*i*m ; 0; 0; ((-1)) *h*i*m ; ((-1)) *d*i*m ; ((-1)) *a*d*e + (1) *a*d*h + ((-1)) *a*e*h + (1) *d*e*h ; ];
[
0; 0; 0; 0; 0; 0; 0; (1) *i ; 0; 0; 0; 0; ((-1)) *i*k ; (1) *i*m ; ((-1)) *i*k + (1) *i*l ; (1) *i*l ; 0; ((-1)) *i*k ; ((-1)) *i^2*k ; ((-1)) *h*i*k + ((-1)) *c*i*m ; ((-1)) *g*i*m ; (1) *e*i*k ; (1) *b*i*m ; (1) *a*e*j + ((-1)) *d*e*j + ((-1)) *a*h*j + (1) *d*h*j + (1) *a*e*k + ((-1)) *d*h*k + ((-1)) *a*i*k + (1) *d*i*k + ((-1)) *e*i*k + (1) *h*i*k + ((-2)) *a*e*l + (1) *d*e*l + (1) *a*h*l ; ];
[
0; 0; 0; 0; 0; ((-1)) *i ; 0; 0; 0; 0; ((-1)) *g ; (1) *g*j ; ((-1)) *i*k + ((-1)) *f*m ; 0; 0; ((-1)) *i*k ; 0; (1) *g*i*k + ((-1)) *i*k^2 ; ((-1)) *i*k^2 ; (1) *c*i*k + ((-1)) *g*i*k ; ((-1)) *g*i*k + (1) *i*k^2 ; ((-1)) *c*i*k + (1) *i*k^2 ; ((-2)) *a*f*g + (1) *d*f*g + (1) *b*c*h + (1) *c*d*k + (1) *b*e*k + (1) *a*f*k + ((-1)) *d*f*k + (1) *a*g*k + ((-1)) *d*g*k + ((-1)) *b*h*k + ((-1)) *c*h*k + ((-1)) *a*k^2 + (1) *d*k^2 + ((-1)) *e*k^2 + (1) *h*k^2 ; ];
[
0; 0; 0; 0; (1) *f ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; ((-1)) *f*g*j ; 0; ((-1)) *c*g*j + (1) *f*g*j ; 0; 0; 0; ];
[
0; 0; (1) *f ; 0; 0; ((-1)) *f ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; (1) *g*j^2 ; 0; 0; 0; ];
[
(1) *a + ((-1)) *d ; 0; 0; (1) *d ; 0; (1) *k + ((-1)) *l ; 0; 0; ((-1)) *d*k + (1) *i*k ; (1) *d*m + (1) *h*m ; ((-1)) *d*k + (1) *d*l ; (1) *d*l ; 0; ((-1)) *a*k + ((-1)) *b*m + (2) *c*m ; 0; ((-1)) *d*h*k + ((-1)) *c*d*m ; ((-1)) *a*i*k + ((-1)) *d*g*m + (1) *c*i*m ; (1) *d*e*k + ((-1)) *h*i*k + (1) *i^2*k ; (1) *b*d*m ; (1) *a*e*j + ((-1)) *d*e*j + ((-1)) *a*h*j + (1) *d*h*j + ((-1)) *a*e*k + (1) *d*h*k ; ];
[
0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; (1) *j ; 0; 0; 0; 0; 0; ((-1)) *d*g + (1) *g*h ; ];
[
0; 0; 0; 0; 0; 0; 0; 0; 0; 0; (1) *j ; 0; 0; (1) *j^2 ; 0; 0; 0; ((-1)) *b*g + (1) *f*g ; ];
[
0; 0; 0; 0; 0; 0; ((-1)) *g ; 0; 0; ((-1)) *c ; 0; 0; ((-1)) *c*j ; 0; 0; 0; 0; ];
[
0; 0; (1) ; 0; 0; ((-1)) *e ; (1) *j ; 0; 0; ((-1)) *c ; 0; (1) *c*i ; 0; 0; ((-1)) *c*i + ((-1)) *a*j + (1) *d*j ; (1) *c*e + (1) *a*g + ((-1)) *d*g + ((-1)) *c*h ; ];
[
0; 0; 0; 0; (1) *j ; 0; 0; 0; 0; (1) *j^2 ; 0; ((-1)) *j^2 ; 0; 0; (1) *b*c + ((-1)) *c*f ; ];
[
0; 0; (1) *e ; 0; (1) *i ; 0; 0; (1) *a ; 0; 0; 0; 0; (1) *a*i ; 0; ];
[
0; (1) *g ; 0; ((-1)) *j ; 0; 0; (1) *c ; (1) *g*i ; 0; 0; 0; 0; 0; ];
[
(1) *j ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; (1) *c*d + ((-1)) *a*f ; ];
[
0; 0; 0; 0; 0; ((-1)) *d ; 0; (1) *a ; (1) *h + ((-1)) *i ; 0; 0; ];
[
0; 0; 0; 0; ((-1)) *b ; 0; (1) *c ; (1) *f ; 0; 0; ];
[
0; 0; 0; 0; ((-1)) *a ; 0; (1) *e ; 0; 0; ];
[
0; 0; ((-1)) *e ; 0; 0; 0; (1) *a ; 0; ];
[
0; 0; ((-1)) *f ; ((-1)) *g ; 0; (1) *b ; 0; ];
[
0; ((-1)) *h + (1) *i ; ((-1)) *e ; 0; (1) *d + ((-1)) *i ; 0; ];
];
c:=
(-1);