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

theorem Th3:
  c + f = ((dom f) --> c) + f
  proof
    set g = (dom f) --> c;
A2: dom(c+f) = dom f by VALUED_1:def 2;
    thus
A3: dom(c+f) = dom g /\ dom f by VALUED_1:def 2
    .= dom(g+f) by VALUED_1:def 1;
    let x be object;
    assume
A4: x in dom(c+f);
    then
A5: g.x = c by A2,FUNCOP_1:7;
    thus (c+f).x = c+f.x by A4,VALUED_1:def 2
    .= (g+f).x by A3,A4,A5,VALUED_1:def 1;
  end;
