reserve A for non empty closed_interval Subset of REAL;

theorem Th23E:
for a,b being Real holds
AffineMap (a,b) = (a (#) ( #Z 1)) + (b (#) ( #Z 0))
proof
 let a,b be Real;
for x being object st x in REAL holds
(AffineMap (a,b)) . x
= ((a (#) ( #Z 1)) + (b (#) ( #Z 0))) . x
proof
  let x be object;
  assume x in REAL; then
  reconsider x as Element of REAL;
  ((a (#) ( #Z 1)) + (b (#) ( #Z 0))) . x
   = (a (#) ( #Z 1)).x  + (b (#) ( #Z 0)) . x by VALUED_1:1
  .= a * ( #Z 1).x  + (b (#) ( #Z 0)) . x by VALUED_1:6
  .= a * ( #Z 1).x  + b * ( #Z 0) . x by VALUED_1:6
  .= a * (x #Z 1)  + b * ( #Z 0) . x by TAYLOR_1:def 1
  .= a * (x #Z 1)  + b * (x #Z 0)  by TAYLOR_1:def 1
  .= a * (x |^ 1)  + b * (x #Z 0)  by PREPOWER:36
  .= a * (x )  + b * (1 ) by PREPOWER:34
  .= (AffineMap(a,b).x) by FCONT_1:def 4;
  hence thesis;
 end;
 hence thesis by FUNCT_2:12;
end;
