let P1, P2, P3, P4, P5, P6, P7, P8, P9 be Point of (ProjectiveSpace (TOP-REAL 3)); :: thesis: for a, b, c, d, e, f being Real st ( not a = 0 or not b = 0 or not c = 0 or not d = 0 or not e = 0 or not f = 0 ) & {P1,P2,P3,P4,P5,P6} c= conic (a,b,c,d,e,f) & P1,P2,P3,P4,P5,P6,P7,P8,P9 are_in_Pascal_configuration holds
P7,P8,P9 are_collinear
let a, b, c, d, e, f be Real; :: thesis: ( ( not a = 0 or not b = 0 or not c = 0 or not d = 0 or not e = 0 or not f = 0 ) & {P1,P2,P3,P4,P5,P6} c= conic (a,b,c,d,e,f) & P1,P2,P3,P4,P5,P6,P7,P8,P9 are_in_Pascal_configuration implies P7,P8,P9 are_collinear )
per cases ( P1,P2,P3 are_collinear or not P1,P2,P3 are_collinear ) ;
suppose P1,P2,P3 are_collinear ; :: thesis: ( ( not a = 0 or not b = 0 or not c = 0 or not d = 0 or not e = 0 or not f = 0 ) & {P1,P2,P3,P4,P5,P6} c= conic (a,b,c,d,e,f) & P1,P2,P3,P4,P5,P6,P7,P8,P9 are_in_Pascal_configuration implies P7,P8,P9 are_collinear )
hence ( ( not a = 0 or not b = 0 or not c = 0 or not d = 0 or not e = 0 or not f = 0 ) & {P1,P2,P3,P4,P5,P6} c= conic (a,b,c,d,e,f) & P1,P2,P3,P4,P5,P6,P7,P8,P9 are_in_Pascal_configuration implies P7,P8,P9 are_collinear ) by Th35; :: thesis: verum
end;
suppose not P1,P2,P3 are_collinear ; :: thesis: ( ( not a = 0 or not b = 0 or not c = 0 or not d = 0 or not e = 0 or not f = 0 ) & {P1,P2,P3,P4,P5,P6} c= conic (a,b,c,d,e,f) & P1,P2,P3,P4,P5,P6,P7,P8,P9 are_in_Pascal_configuration implies P7,P8,P9 are_collinear )
hence ( ( not a = 0 or not b = 0 or not c = 0 or not d = 0 or not e = 0 or not f = 0 ) & {P1,P2,P3,P4,P5,P6} c= conic (a,b,c,d,e,f) & P1,P2,P3,P4,P5,P6,P7,P8,P9 are_in_Pascal_configuration implies P7,P8,P9 are_collinear ) by Th34; :: thesis: verum
end;
end;