reserve c for Complex;
reserve r for Real;
reserve m,n for Nat;
reserve f for complex-valued Function;
reserve f,g for differentiable Function of REAL,REAL;

theorem Th19:
  (r(#)f)`| = r(#)(f`|)
  proof
    let s be Element of REAL;
A1: f is_differentiable_in s by Th9;
A2: f`|.s = diff(f,s) by Th10;
    thus (r(#)f)`|.s = diff(r(#)f,s) by Th10
    .= r*diff(f,s) by A1,FDIFF_1:15
    .= (r(#)(f`|)).s by A2,VALUED_1:6;
  end;
