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 Th27:
  (f1(#)f2)^ = (f1^)(#)(f2^)
proof
A1: dom ((f1(#)f2)^) = dom (f1(#)f2) \ (f1(#)f2)"{0} by Def2
    .= (dom f1 \ f1"{0}) /\ (dom f2 \ (f2)"{0}) by Th2
    .= dom (f1^) /\ (dom f2 \ (f2)"{0}) by Def2
    .= dom (f1^) /\ dom (f2^) by Def2
    .= dom ((f1^) (#) (f2^)) by VALUED_1:def 4;
  now
    let c be object;
    assume
A2: c in dom ((f1(#)f2)^);
    then
A3: c in dom (f1^) /\ dom (f2^) by A1,VALUED_1:def 4;
    then
A4: c in dom (f1^) by XBOOLE_0:def 4;
A5: c in dom (f2^) by A3,XBOOLE_0:def 4;
    thus ((f1(#)f2)^).c = ((f1(#)f2).c)" by A2,Def2
      .= (f1.c * f2.c)" by VALUED_1:5
      .= (f1.c)"* (f2.c)" by XCMPLX_1:204
      .= ((f1^).c)*(f2.c)" by A4,Def2
      .= ((f1^).c) *((f2^).c) by A5,Def2
      .= ((f1^) (#) (f2^)).c by VALUED_1:5;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
