reserve T,T1,T2,S for non empty TopSpace;
reserve p,q for Point of TOP-REAL 2;

theorem Th36:
  for K0,B0 being Subset of TOP-REAL 2,f being Function of (
TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st f=Out_In_Sq|K0 & B0=NonZero TOP-REAL 2 & K0={
p: (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2} holds f is
  continuous
proof
  let K0,B0 be Subset of TOP-REAL 2,f be Function of (TOP-REAL 2)|K0,(TOP-REAL
  2)|B0;
A1: (1.REAL 2)<>0.TOP-REAL 2 by Lm1,REVROT_1:19;
  assume
A2: f=Out_In_Sq|K0 & B0=NonZero TOP-REAL 2 & K0={p:(p`2<=p`1 & -p`1<=p`2
  or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2};
A3: K0 c= B0
  proof
    let x be object;
    assume
A4: x in K0;
    then
    ex p8 being Point of TOP-REAL 2 st x=p8 &( p8`2<=p8`1 & - p8`1<=p8`2 or
    p8`2>=p8`1 & p8`2<=-p8`1)& p8<>0.TOP-REAL 2 by A2;
    then not x in {0.TOP-REAL 2} by TARSKI:def 1;
    hence thesis by A2,A4,XBOOLE_0:def 5;
  end;
  (1.REAL 2)`2<=(1.REAL 2)`1 & -(1.REAL 2)`1<=(1.REAL 2)`2 or (1.REAL 2)
  `2>=(1.REAL 2)`1 & (1.REAL 2)`2<=-(1.REAL 2)`1 by Th5;
  then
A5: 1.REAL 2 in K0 by A2,A1;
  then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
A6: K1 c= NonZero TOP-REAL 2
  proof
    let z be object;
    assume
A7: z in K1;
    then ex p8 being Point of TOP-REAL 2 st p8=z &( p8`2<=p8`1 & - p8`1<=p8`2
    or p8`2>=p8`1 & p8`2<=-p8`1)& p8<>0.TOP-REAL 2 by A2;
    then not z in {0.TOP-REAL 2} by TARSKI:def 1;
    hence thesis by A7,XBOOLE_0:def 5;
  end;
A8: dom (Out_In_Sq|K1) c= dom ((proj2)*(Out_In_Sq|K1))
  proof
    let x be object;
    assume
A9: x in dom (Out_In_Sq|K1);
    then x in dom Out_In_Sq /\ K1 by RELAT_1:61;
    then x in dom Out_In_Sq by XBOOLE_0:def 4;
    then Out_In_Sq.x in rng Out_In_Sq by FUNCT_1:3;
    then
A10: dom proj2 = (the carrier of TOP-REAL 2) & Out_In_Sq.x in the carrier
    of TOP-REAL 2 by FUNCT_2:def 1,XBOOLE_0:def 5;
    (Out_In_Sq|K1).x=Out_In_Sq.x by A9,FUNCT_1:47;
    hence thesis by A9,A10,FUNCT_1:11;
  end;
A11: rng ((proj2)*(Out_In_Sq|K1)) c= the carrier of R^1 by TOPMETR:17;
A12: NonZero TOP-REAL 2<>{} by Th9;
A13: dom (Out_In_Sq|K1) c= dom ((proj1)*(Out_In_Sq|K1))
  proof
    let x be object;
    assume
A14: x in dom (Out_In_Sq|K1);
    then x in dom Out_In_Sq /\ K1 by RELAT_1:61;
    then x in dom Out_In_Sq by XBOOLE_0:def 4;
    then Out_In_Sq.x in rng Out_In_Sq by FUNCT_1:3;
    then
A15: dom proj1 = (the carrier of TOP-REAL 2) & Out_In_Sq.x in the carrier
    of TOP-REAL 2 by FUNCT_2:def 1,XBOOLE_0:def 5;
    (Out_In_Sq|K1).x=Out_In_Sq.x by A14,FUNCT_1:47;
    hence thesis by A14,A15,FUNCT_1:11;
  end;
A16: rng ((proj1)*(Out_In_Sq|K1)) c= the carrier of R^1 by TOPMETR:17;
  dom ((proj1)*(Out_In_Sq|K1)) c= dom (Out_In_Sq|K1) by RELAT_1:25;
  then dom ((proj1)*(Out_In_Sq|K1)) =dom (Out_In_Sq|K1) by A13
    .=dom Out_In_Sq /\ K1 by RELAT_1:61
    .=(NonZero TOP-REAL 2)/\ K1 by A12,FUNCT_2:def 1
    .=K1 by A6,XBOOLE_1:28
    .=[#]((TOP-REAL 2)|K1) by PRE_TOPC:def 5
    .=the carrier of (TOP-REAL 2)|K1;
  then reconsider
  g1=(proj1)*(Out_In_Sq|K1) as Function of (TOP-REAL 2)|K1,R^1 by A16,FUNCT_2:2
;
  for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1
  holds g1.p=1/p`1
  proof
A17: K1 c= NonZero TOP-REAL 2
    proof
      let z be object;
      assume
A18:  z in K1;
      then
      ex p8 being Point of TOP-REAL 2 st p8=z &( p8`2<=p8`1 & - p8`1<=p8`2
      or p8`2>=p8`1 & p8`2<=-p8`1)& p8<>0.TOP-REAL 2 by A2;
      then not z in {0.TOP-REAL 2} by TARSKI:def 1;
      hence thesis by A18,XBOOLE_0:def 5;
    end;
A19: NonZero TOP-REAL 2<>{} by Th9;
A20: dom (Out_In_Sq|K1)=dom Out_In_Sq /\ K1 by RELAT_1:61
      .=(NonZero TOP-REAL 2)/\ K1 by A19,FUNCT_2:def 1
      .=K1 by A17,XBOOLE_1:28;
    let p be Point of TOP-REAL 2;
A21: the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1)
      .=K1 by PRE_TOPC:def 5;
    assume
A22: p in the carrier of (TOP-REAL 2)|K1;
    then ex p3 being Point of TOP-REAL 2 st p=p3 &( p3`2<=p3`1 & - p3`1<=p3`2
    or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A2,A21;
    then
A23: Out_In_Sq.p=|[1/p`1,p`2/p`1/p`1]| by Def1;
    (Out_In_Sq|K1).p=Out_In_Sq.p by A22,A21,FUNCT_1:49;
    then g1.p=(proj1).(|[1/p`1,p`2/p`1/p`1]|) by A22,A20,A21,A23,FUNCT_1:13
      .=(|[1/p`1,p`2/p`1/p`1]|)`1 by PSCOMP_1:def 5
      .=1/p`1 by EUCLID:52;
    hence thesis;
  end;
  then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A24: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
  |K1 holds f1.p=1/p`1;
  dom ((proj2)*(Out_In_Sq|K1)) c= dom (Out_In_Sq|K1) by RELAT_1:25;
  then dom ((proj2)*(Out_In_Sq|K1)) =dom (Out_In_Sq|K1) by A8
    .=dom Out_In_Sq /\ K1 by RELAT_1:61
    .=(NonZero TOP-REAL 2)/\ K1 by A12,FUNCT_2:def 1
    .=K1 by A6,XBOOLE_1:28
    .=[#]((TOP-REAL 2)|K1) by PRE_TOPC:def 5
    .=the carrier of (TOP-REAL 2)|K1;
  then reconsider
  g2=(proj2)*(Out_In_Sq|K1) as Function of (TOP-REAL 2)|K1,R^1 by A11,FUNCT_2:2
;
  for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1
  holds g2.p=p`2/p`1/p`1
  proof
A25: NonZero TOP-REAL 2<>{} by Th9;
A26: dom (Out_In_Sq|K1)=dom Out_In_Sq /\ K1 by RELAT_1:61
      .=(NonZero TOP-REAL 2)/\ K1 by A25,FUNCT_2:def 1
      .=K1 by A6,XBOOLE_1:28;
    let p be Point of TOP-REAL 2;
A27: the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1)
      .=K1 by PRE_TOPC:def 5;
    assume
A28: p in the carrier of (TOP-REAL 2)|K1;
    then ex p3 being Point of TOP-REAL 2 st p=p3 &( p3`2<=p3`1 & - p3`1<=p3`2
    or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A2,A27;
    then
A29: Out_In_Sq.p=|[1/p`1,p`2/p`1/p`1]| by Def1;
    (Out_In_Sq|K1).p=Out_In_Sq.p by A28,A27,FUNCT_1:49;
    then g2.p=(proj2).(|[1/p`1,p`2/p`1/p`1]|) by A28,A26,A27,A29,FUNCT_1:13
      .=(|[1/p`1,p`2/p`1/p`1]|)`2 by PSCOMP_1:def 6
      .=p`2/p`1/p`1 by EUCLID:52;
    hence thesis;
  end;
  then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A30: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
  |K1 holds f2.p=p`2/p`1/p`1;
A31: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
  holds q`1<>0
  proof
    let q be Point of TOP-REAL 2;
A32: the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1)
      .=K1 by PRE_TOPC:def 5;
    assume q in the carrier of (TOP-REAL 2)|K1;
    then
A33: ex p3 being Point of TOP-REAL 2 st q=p3 &( p3`2<=p3`1 & - p3`1<=p3`2
    or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A2,A32;
    now
      assume
A34:  q`1=0;
      then q`2=0 by A33;
      hence contradiction by A33,A34,EUCLID:53,54;
    end;
    hence thesis;
  end;
  then
A35: f1 is continuous by A24,Th31;
A36: for x,y,r,s being Real st |[x,y]| in K1 & r=f1.(|[x,y]|) & s=f2.
  (|[x,y]|) holds f. |[x,y]|=|[r,s]|
  proof
    let x,y,r,s be Real;
    assume that
A37: |[x,y]| in K1 and
A38: r=f1.(|[x,y]|) & s=f2.(|[x,y]|);
    set p99=|[x,y]|;
A39: the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1)
      .=K1 by PRE_TOPC:def 5;
    then
A40: f1.p99=1/p99`1 by A24,A37;
A41: ex p3 being Point of TOP-REAL 2 st p99=p3 &( p3`2<=p3`1 & -p3`1<=p3`2
    or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A2,A37;
    then (p99`2<=p99`1 & -p99`1<=p99`2 or p99`2>=p99`1 & p99`2<=-p99`1)
    implies Out_In_Sq.p99=|[1/p99`1,p99`2/p99`1/p99`1]| by Def1;
    then
(Out_In_Sq|K0). |[x,y]|= |[1/p99`1,p99`2/p99`1/p99`1]| by A37,A41,FUNCT_1:49
      .=|[r,s]| by A30,A37,A38,A39,A40;
    hence thesis by A2;
  end;
  f2 is continuous by A31,A30,Th33;
  hence thesis by A5,A3,A35,A36,Th35;
end;
