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
  (f(#)g)`| = g(#)(f`|) + 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;
A3: (g.s)*(f`|.s) = (g(#)(f`|)).s & (f.s)*(g`|.s) = (f(#)(g`|)).s
    by VALUED_1:5;
    thus (f(#)g)`|.s = diff(f(#)g,s) by Th10
    .= (g.s)*diff(f,s) + (f.s)*diff(g,s) by A1,FDIFF_1:16
    .= (g(#)(f`|) + f(#)(g`|)).s by A2,A3,VALUED_1:1;
  end;
