let x0, x1, x2, a, b be Real; :: thesis: ( x0,x1,x2 are_mutually_distinct implies [!(AffineMap (a,b)),x0,x1,x2!] = 0 )
assume A1: x0,x1,x2 are_mutually_distinct ; :: thesis: [!(AffineMap (a,b)),x0,x1,x2!] = 0
then A2: x1 <> x2 by ZFMISC_1:def 5;
x0 <> x1 by A1, ZFMISC_1:def 5;
then [!(AffineMap (a,b)),x0,x1,x2!] = (a - [!(AffineMap (a,b)),x1,x2!]) / (x0 - x2) by Th21
.= (a - a) / (x0 - x2) by A2, Th21 ;
hence [!(AffineMap (a,b)),x0,x1,x2!] = 0 ; :: thesis: verum