reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;

theorem
  (r*p)(#)f = r(#)(p(#)f)
proof
A1: dom ((r*p) (#) f) = dom f by VALUED_1:def 5
    .= dom (p(#)f) by VALUED_1:def 5
    .= dom (r(#)(p(#)f)) by VALUED_1:def 5;
  now
    let c be object;
    assume
A2: c in dom ((r*p)(#)f);
    then
A3: c in dom (p(#)f) by A1,VALUED_1:def 5;
    thus ((r*p)(#)f).c = r*p * f.c by A2,VALUED_1:def 5
      .= r*(p * f.c)
      .= r * (p(#)f).c by A3,VALUED_1:def 5
      .= (r(#)(p(#)f)).c by A1,A2,VALUED_1:def 5;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
