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 Th15:
  (f-g)`| = f`| - g`|
  proof
    let s be Element of REAL;
A1: f is_differentiable_in s & g is_differentiable_in s by Th9;
A2: f`|.s = diff(f,s) & g`|.s = diff(g,s) by Th10;
    thus (f-g)`|.s = diff(f-g,s) by Th10
    .= diff(f,s) - diff(g,s) by A1,FDIFF_1:14
    .= (f`| - g`|).s by A2,VALUED_1:15;
  end;
