reserve A for non empty closed_interval Subset of REAL;

theorem Th23C:
for a,b being Real holds
(id REAL) (#) (AffineMap (a,b)) = (a (#) ( #Z 2)) + (b (#) ( #Z 1))
proof
 let a,b be Real;
for x being object st x in REAL holds
((id REAL) (#) AffineMap (a,b)) . x
= ((a (#) ( #Z 2)) + (b (#) ( #Z 1))) . x
proof
  let x be object;
  assume x in REAL;then
  reconsider x as Element of REAL;
  ((a (#) ( #Z 2)) + (b (#) ( #Z 1))) . x
   = (a (#) ( #Z 2)).x  + (b (#) ( #Z 1)) . x by VALUED_1:1
  .= a * ( #Z 2).x + (b (#) ( #Z 1)) . x by VALUED_1:6
  .= a * ( #Z 2).x + b * ( #Z 1) . x by VALUED_1:6
  .= a * (x #Z 2) + b * ( #Z 1) . x by TAYLOR_1:def 1
  .= a * (x #Z 2) + b * (x #Z 1)  by TAYLOR_1:def 1
  .= a * (x |^ 2) + b * (x #Z 1)  by PREPOWER:36
  .= a * (x |^ 2) + b * (x )  by PREPOWER:35
  .= a * (x * x) + b * (x ) by NEWTON:81
  .= x * (a * x + b)
  .= ((id REAL).x) * (AffineMap(a,b).x) by FCONT_1:def 4;
  hence thesis by VALUED_1:5;
 end;
 hence thesis by FUNCT_2:12;
end;
