reserve h,r,r1,r2,x0,x1,x2,x3,x4,x5,x,a,b,c,k for Real,
  f,f1,f2 for Function of REAL,REAL;

theorem
  for x holds fD(AffineMap(a,b),h).x = a*h
proof
  let x;
  fD(AffineMap(a,b),h).x = AffineMap(a,b).(x+h)-AffineMap(a,b).x by DIFF_1:3
    .= a*(x+h)+b-AffineMap(a,b).x by FCONT_1:def 4
    .= a*x+a*h+b-(a*x+b) by FCONT_1:def 4;
  hence thesis;
end;
