 reserve x,y for Element of [.0,1.];

theorem LemmaAffine:
  for a,b being Real holds
    AffineMap (a,b) is_differentiable_on REAL &
   for x being Real holds diff (AffineMap (a,b), x) = a
  proof
    let a,b be Real;
    reconsider Z = [#]REAL as open Subset of REAL;
A1: dom AffineMap (a,b) = REAL by FUNCT_2:def 1;
A2: for x being Real st x in Z holds
      AffineMap (a,b).x = a * x + b by FCONT_1:def 4;
    hence
Ka: AffineMap (a,b) is_differentiable_on REAL by A1,FDIFF_1:23;
    let x be Real;
AC: x in Z by XREAL_0:def 1;
    ((AffineMap(a,b))`|Z).x = a by A1,A2,FDIFF_1:23,XREAL_0:def 1;
    hence thesis by AC,FDIFF_1:def 7,Ka;
  end;
