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 Th34:
  (f/g)(#)(f1/g1) = (f(#)f1)/(g(#)g1)
proof
A1: now
    let c be object;
    assume c in dom ((f/g)(#)(f1/g1));
    thus ((f/g)(#)(f1/g1)).c = ((f/g).c)* (f1/g1).c by VALUED_1:5
      .= ((f(#)(g^)).c) * (f1/g1).c by Th31
      .= ((f(#)(g^)).c) * (f1(#)(g1^)).c by Th31
      .= (f.c) * ((g^).c) * (f1(#)(g1^)).c by VALUED_1:5
      .= (f.c) * ((g^).c) * ((f1.c)* (g1^).c) by VALUED_1:5
      .= (f.c) * ((f1.c) * (((g^).c) * (g1^).c))
      .= (f.c) * ((f1.c) * ((g^)(#)(g1^)).c) by VALUED_1:5
      .= (f.c) * (f1.c) * ((g^)(#)(g1^)).c
      .= (f.c) * (f1.c) * ((g(#)g1)^).c by Th27
      .= ((f(#)f1).c) * ((g(#)g1)^).c by VALUED_1:5
      .= ((f(#)f1)(#)((g(#)g1)^)).c by VALUED_1:5
      .= ((f(#)f1)/(g(#)g1)).c by Th31;
  end;
  dom ((f/g)(#)(f1/g1)) = dom (f/g) /\ dom (f1/g1) by VALUED_1:def 4
    .= dom f /\ (dom g \ g"{0}) /\ dom (f1/g1) by Def1
    .= dom f /\ (dom g \ g"{0}) /\ (dom f1 /\ (dom g1 \ g1"{0})) by Def1
    .= dom f /\ ((dom g \ g"{0}) /\ (dom f1 /\ (dom g1 \ g1"{0}))) by
XBOOLE_1:16
    .= dom f /\ (dom f1 /\ ((dom g \ g"{0}) /\ (dom g1 \ g1"{0}))) by
XBOOLE_1:16
    .= dom f /\ dom f1 /\ ((dom g \ g"{0}) /\ (dom g1 \ g1"{0})) by XBOOLE_1:16
    .= dom (f(#)f1) /\ ((dom g \ g"{0}) /\ (dom g1 \ g1"{0})) by VALUED_1:def 4
    .= dom (f(#)f1) /\ (dom (g(#)g1) \ (g(#)g1)"{0}) by Th2
    .= dom ((f(#)f1)/(g(#)g1)) by Def1;
  hence thesis by A1,FUNCT_1:2;
end;
