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

theorem Th4:
  f - c = f - ((dom f) --> c)
  proof
    set g = (dom f) --> c;
A2: dom(f-c) = dom f by VALUED_1:3;
    thus
A3: dom(f-c) = dom g /\ dom f by VALUED_1:3
    .= dom(f-g) by VALUED_1:12;
    let x be object;
    assume
A4: x in dom(f-c);
    then
A5: g.x = c by A2,FUNCOP_1:7;
    thus (f-c).x = f.x-c by A3,A4,VALUED_1:3
    .= (f-g).x by A3,A4,A5,VALUED_1:13;
  end;
