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

theorem
  f - 0 = f
  proof
    thus dom(f-0) = dom f by VALUED_1:3;
    let x be object;
    assume
A1: x in dom (f-0);
    dom(f-0) = dom f by VALUED_1:3;
    hence (f-0).x = (f.x)-0 by A1,VALUED_1:3
    .= f.x;
  end;
