reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th41:
  (f/g)(#)(f1/g1) = (f(#)f1)/(g(#)g1)
proof
A1: now
    let c;
    assume
A2: c in dom ((f/g)(#)(f1/g1));
    then c in dom (f/g) /\ dom (f1/g1) by Th3;
    then
A3: c in dom (f (#)(g^)) /\ dom (f1/g1) by Th38;
    then
A4: c in dom (f (#)(g^)) by XBOOLE_0:def 4;
    then
A5: c in dom f /\ dom(g^) by Th3;
    c in dom (f (#)(g^)) /\ dom (f1(#)(g1^)) by A3,Th38;
    then
A6: c in dom (f1(#)(g1^)) by XBOOLE_0:def 4;
    then
A7: c in dom f1 /\ dom(g1^) by Th3;
    then
A8: c in dom f1 by XBOOLE_0:def 4;
    c in dom f by A5,XBOOLE_0:def 4;
    then c in dom f /\ dom f1 by A8,XBOOLE_0:def 4;
    then
A9: c in dom (f(#)f1) by Th3;
A10: c in dom(g1^) by A7,XBOOLE_0:def 4;
    c in dom(g^) by A5,XBOOLE_0:def 4;
    then c in dom (g^) /\ dom (g1^) by A10,XBOOLE_0:def 4;
    then
A11: c in dom((g^)(#)(g1^)) by Th3;
    then c in dom((g(#)g1)^) by Th34;
    then c in dom (f(#)f1) /\ dom((g(#)g1)^) by A9,XBOOLE_0:def 4;
    then
A12: c in dom ((f(#)f1)(#)((g(#)g1)^)) by Th3;
    thus ((f/g)(#)(f1/g1))/.c = ((f/g)/.c)* ((f1/g1)/.c) by A2,Th3
      .= ((f(#)(g^))/.c) * ((f1/g1)/.c) by Th38
      .= ((f(#)(g^))/.c) * ((f1(#)(g1^))/.c) by Th38
      .= ((f/.c)) * ((g^)/.c) *((f1(#)(g1^))/.c) by A4,Th3
      .= ((f/.c)) * ((g^)/.c) * ((((f1/.c)))* ((g1^)/.c)) by A6,Th3
      .= ((f/.c)) * ((((f1/.c))) * (((g^)/.c) * ((g1^)/.c)))
      .= ((f/.c)) * ((((f1/.c))) * (((g^)(#)(g1^))/.c)) by A11,Th3
      .= ((f/.c)) * (((f1/.c))) * (((g^)(#)(g1^))/.c)
      .= ((f/.c)) * (((f1/.c))) * (((g(#)g1)^)/.c) by Th34
      .= ((f(#)f1)/.c) * (((g(#)g1)^)/.c) by A9,Th3
      .= ((f(#)f1)(#)((g(#)g1)^))/.c by A12,Th3
      .= ((f(#)f1)/(g(#)g1))/.c by Th38;
  end;
  dom ((f/g)(#)(f1/g1)) = dom (f/g) /\ dom (f1/g1) by Th3
    .= dom f /\ (dom g \ g"{0c}) /\ dom (f1/g1) by Def1
    .= dom f /\ (dom g \ g"{0c}) /\ (dom f1 /\ (dom g1 \ g1"{0c})) by Def1
    .= dom f /\ ((dom g \ g"{0c}) /\ (dom f1 /\ (dom g1 \ g1"{0c}))) by
XBOOLE_1:16
    .= dom f /\ (dom f1 /\ ((dom g \ g"{0c}) /\ (dom g1 \ g1"{0c}))) by
XBOOLE_1:16
    .= dom f /\ dom f1 /\ ((dom g \ g"{0c}) /\ (dom g1 \ g1"{0c})) by
XBOOLE_1:16
    .= dom (f(#)f1) /\ ((dom g \ g"{0c}) /\ (dom g1 \ g1"{0c})) by Th3
    .= dom (f(#)f1) /\ (dom (g(#)g1) \ (g(#)g1)"{0c}) by Th7
    .= dom ((f(#)f1)/(g(#)g1)) by Def1;
  hence thesis by A1,PARTFUN2:1;
end;
