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