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 Th47:
  f1/f + g1/g = (f1(#)g + g1(#)f)/(f(#)g)
proof
A1: now
    let c;
A2: dom (f^) c= dom f by Th6;
    assume
A3: c in dom ((f1/f) + (g1/g));
    then
A4: c in dom (f1/f) /\ dom (g1/g) by VALUED_1:def 1;
    then
A5: c in dom (f1/f) by XBOOLE_0:def 4;
A6: c in dom (g1/g) by A4,XBOOLE_0:def 4;
A7: c in dom (f1 (#)(f^)) /\ dom (g1/g) by A4,Th38;
    then c in dom (f1 (#)(f^)) by XBOOLE_0:def 4;
    then
A8: c in dom f1 /\ dom(f^) by Th3;
    then
A9: c in dom(f^) by XBOOLE_0:def 4;
    then
A10: (f/.c) <> 0c by Th8;
    c in dom (f1 (#)(f^)) /\ dom (g1(#)(g^)) by A7,Th38;
    then c in dom (g1(#)(g^)) by XBOOLE_0:def 4;
    then
A11: c in dom g1 /\ dom(g^) by Th3;
    then
A12: c in dom(g^) by XBOOLE_0:def 4;
    then
A13: g/.c <> 0c by Th8;
    c in dom g1 by A11,XBOOLE_0:def 4;
    then c in dom g1 /\ dom f by A9,A2,XBOOLE_0:def 4;
    then
A14: c in dom (g1(#)f) by Th3;
A15: dom (g^) c= dom g by Th6;
    then c in dom f /\ dom g by A9,A12,A2,XBOOLE_0:def 4;
    then
A16: c in dom (f(#)g) by Th3;
    c in dom f1 by A8,XBOOLE_0:def 4;
    then c in dom f1 /\ dom g by A12,A15,XBOOLE_0:def 4;
    then
A17: c in dom (f1(#)g) by Th3;
    then c in dom (f1(#)g) /\ dom (g1(#)f) by A14,XBOOLE_0:def 4;
    then
A18: c in dom (f1(#)g + g1(#)f) by VALUED_1:def 1;
    c in dom (f^) /\ dom (g^) by A9,A12,XBOOLE_0:def 4;
    then c in dom ((f^)(#)(g^)) by Th3;
    then c in dom ((f(#)g)^) by Th34;
    then c in dom (f1(#)g + g1(#)f) /\ dom ((f(#)g)^) by A18,XBOOLE_0:def 4;
    then c in dom ((f1(#)g + g1(#)f)(#)((f(#)g)^)) by Th3;
    then
A19: c in dom ((f1(#)g + g1(#)f)/(f(#)g)) by Th38;
    thus (f1/f + g1/g)/.c = (f1/f)/.c + (g1/g)/.c by A3,Th1
      .= (((f1/.c))) * ((f/.c))" + (g1/g)/.c by A5,Def1
      .= (((f1/.c))) *(1r*((f/.c))") + (g1/.c) * 1r * (g/.c)" by A6,Def1,
COMPLEX1:def 4
      .= (((f1/.c))) *((g/.c) *(g/.c)"* ((f/.c))") + (g1/.c) * (1r * (g/.c)"
    ) by A13,COMPLEX1:def 4,XCMPLX_0:def 7
      .= (((f1/.c))) *(g/.c *((g/.c)"* ((f/.c))")) + (g1/.c) * (((f/.c)) *((
    f/.c))" * (g/.c)") by A10,COMPLEX1:def 4,XCMPLX_0:def 7
      .= (((f1/.c))) *((g/.c) *((g/.c * (f/.c))")) + (g1/.c) * (((f/.c)) *((
    (f/.c))" * (g/.c)")) by XCMPLX_1:204
      .= (((f1/.c))) *((g/.c) *(((f/.c)* (g/.c))")) + (g1/.c) * (((f/.c)) *(
    ((f/.c) * (g/.c))")) by XCMPLX_1:204
      .= (((f1/.c))) *((g/.c) * ((f(#)g)/.c)") + (g1/.c) * (((f/.c)) *(((f/.
    c) * (g/.c))")) by A16,Th3
      .= (((f1/.c))) *(g/.c) * ((f(#)g)/.c)" + (g1/.c) * (((f/.c)) * ((f(#)
    g)/.c)") by A16,Th3
      .= (f1(#)g)/.c * ((f(#)g)/.c)" + (g1/.c) *(f/.c) *((f(#)g)/.c)" by A17
,Th3
      .= (f1(#)g)/.c * ((f(#)g)/.c)" + (g1(#)f)/.c *((f(#)g)/.c)" by A14,Th3
      .= ((f1(#)g)/.c + (g1(#)f)/.c) *((f(#)g)/.c)"
      .= (f1(#)g + g1(#)f)/.c *((f(#)g)/.c)" by A18,Th1
      .= ((f1(#)g + g1(#)f)/(f(#)g))/.c by A19,Def1;
  end;
  dom ((f1/f) + (g1/g)) = dom (f1/f) /\ dom (g1/g) by VALUED_1:def 1
    .= dom f1 /\ (dom f \ f"{0c}) /\ dom (g1/g) by Def1
    .= dom f1 /\ (dom f \ f"{0c}) /\ (dom g1 /\ (dom g \ g"{0c})) by Def1
    .= dom f1 /\ (dom f /\ (dom f \ f"{0c})) /\ (dom g1 /\ (dom g \ g"{0c}))
  by Th6
    .= dom f /\ (dom f \ f"{0c}) /\ dom f1 /\ (dom g /\ (dom g \ g"{0c}) /\
  dom g1) by Th6
    .= dom f /\ (dom f \ f"{0c}) /\ (dom f1 /\ (dom g /\ (dom g \ g"{0c}) /\
  dom g1)) by XBOOLE_1:16
    .= dom f /\ (dom f \ f"{0c}) /\ (dom f1 /\ (dom g /\ (dom g \ g"{0c}))
  /\ dom g1) by XBOOLE_1:16
    .= dom f /\ (dom f \ f"{0c}) /\ (dom f1 /\ dom g /\ (dom g \ g"{0c}) /\
  dom g1) by XBOOLE_1:16
    .= dom f /\ (dom f \ f"{0c}) /\ (dom (f1(#) g) /\ (dom g \ g"{0c}) /\
  dom g1) by Th3
    .= dom f /\ (dom f \ f"{0c}) /\ (dom (f1(#)g) /\ (dom g1 /\ (dom g \ g"{
  0c}))) by XBOOLE_1:16
    .= dom (f1(#)g) /\ ((dom f \ f"{0c}) /\ dom f /\ (dom g1 /\ (dom g \ g"{
  0c}))) by XBOOLE_1:16
    .= dom (f1(#)g) /\ ((dom f \ f"{0c}) /\ (dom f /\ (dom g1 /\ (dom g \ g"
  {0c})))) by XBOOLE_1:16
    .= dom (f1(#)g) /\ ((dom f \ f"{0c}) /\ (dom g1 /\ dom f /\ (dom g \ g"{
  0c}))) by XBOOLE_1:16
    .= dom (f1(#)g) /\ ((dom f \ f"{0c}) /\ (dom (g1(#)f) /\ (dom g \ g"{0c}
  ))) by Th3
    .= dom (f1(#)g) /\ (dom (g1(#)f) /\ ((dom f \ f"{0c}) /\ (dom g \ g"{0c}
  ))) by XBOOLE_1:16
    .= dom (f1(#)g) /\ dom (g1(#)f) /\ ((dom f \ f"{0c}) /\ (dom g \ g"{0c})
  ) by XBOOLE_1:16
    .= dom (f1(#)g + g1(#)f) /\ ((dom f \ f"{0c}) /\ (dom g \ g"{0c})) by
VALUED_1:def 1
    .= dom (f1(#)g + g1(#)f) /\ (dom (f(#)g) \ (f(#)g)"{0c}) by Th7
    .= dom ((f1(#)g + g1(#)f)/(f(#)g)) by Def1;
  hence thesis by A1,PARTFUN2:1;
end;
