 reserve A for non empty Subset of REAL;

theorem Th28:
  for a,b,r being Real holds
    r (#) (AffineMap (a,b)) = AffineMap (r*a,r*b)
proof
  let a,b,r be Real;
  D1: dom(r (#) (AffineMap (a,b))) = REAL by FUNCT_2:def 1;
  D2: REAL = dom( AffineMap (r*a,r*b)) by FUNCT_2:def 1;
 for x being object st x in dom( AffineMap (r*a,r*b)) holds
 (r (#) (AffineMap (a,b)) ).x = AffineMap (r*a,r*b).x
 proof
  let x be object;
  assume x in dom( AffineMap (r*a,r*b)); then
  reconsider x as Real;
  (r (#) (AffineMap (a,b)) ).x
   = r * (AffineMap (a,b)).x by VALUED_1:6
  .= r * (a*x+b) by FCONT_1:def 4
  .= (r * a)*x+(r*b)
  .= AffineMap (r*a,r*b).x by FCONT_1:def 4;
  hence thesis;
 end;
 hence thesis by FUNCT_1:2,D1,D2;
end;
