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

theorem Th5:
  c (#) f = ((dom f) --> c) (#) f
  proof
    set g = (dom f) --> c;
A2: dom(c(#)f) = dom f by VALUED_1:def 5;
    thus
A3: dom(c(#)f) = dom g /\ dom f by VALUED_1:def 5
    .= dom(g(#)f) by VALUED_1:def 4;
    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 VALUED_1:6
    .= (g(#)f).x by A3,A4,A5,VALUED_1:def 4;
  end;
