let x0, x1, x2, x3, a, b be Real; :: thesis: ( x0,x1,x2,x3 are_mutually_distinct implies [!(AffineMap (a,b)),x0,x1,x2,x3!] = 0 )
assume A1: x0,x1,x2,x3 are_mutually_distinct ; :: thesis: [!(AffineMap (a,b)),x0,x1,x2,x3!] = 0
then A2: x1 <> x2 by ZFMISC_1:def 6;
( x1 <> x3 & x2 <> x3 ) by A1, ZFMISC_1:def 6;
then A3: x1,x2,x3 are_mutually_distinct by A2, ZFMISC_1:def 5;
( x0 <> x1 & x0 <> x2 ) by A1, ZFMISC_1:def 6;
then x0,x1,x2 are_mutually_distinct by A2, ZFMISC_1:def 5;
then [!(AffineMap (a,b)),x0,x1,x2,x3!] = (0 - [!(AffineMap (a,b)),x1,x2,x3!]) / (x0 - x3) by Th22
.= (0 - 0) / (x0 - x3) by A3, Th22 ;
hence [!(AffineMap (a,b)),x0,x1,x2,x3!] = 0 ; :: thesis: verum