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

theorem Th41:
  for B,K0,Kb being Subset of TOP-REAL 2 st B={0.TOP-REAL 2} & K0=
{p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1} & Kb={q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=
1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1}
  ex f being Function of (TOP-REAL 2)|B`,(TOP-REAL 2)|B` st f is continuous & f
  is one-to-one & (for t being Point of TOP-REAL 2 st t in K0 & t<>0.TOP-REAL 2
holds not f.t in K0 \/ Kb) &(for r being Point of TOP-REAL 2 st not r in K0 \/
  Kb holds f.r in K0) & for s being Point of TOP-REAL 2 st s in Kb holds f.s=s
proof
  set K1a={p8 where p8 is Point of TOP-REAL 2: (p8`1<=p8`2 & -p8`2<=p8`1 or p8
  `1>=p8`2 & p8`1<=-p8`2) & p8<>0.TOP-REAL 2 };
  set K0a={p8 where p8 is Point of TOP-REAL 2: (p8`2<=p8`1 & -p8`1<=p8`2 or p8
  `2>=p8`1 & p8`2<=-p8`1) & p8<>0.TOP-REAL 2};
  let B,K0,Kb be Subset of TOP-REAL 2;
  assume
A1: B={0.TOP-REAL 2} & K0={p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1} & Kb={q:
-1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1
  <=1 or 1=q`2 & -1<=q`1 & q`1<=1};
  then reconsider D=B` as non empty Subset of TOP-REAL 2 by Th9;
A2: D`={0.TOP-REAL 2} by A1;
A3: B`=NonZero TOP-REAL 2 by A1,SUBSET_1:def 4;
A4: for t being Point of TOP-REAL 2 st t in K0 & t<>0.TOP-REAL 2 holds not
  Out_In_Sq.t in K0 \/ Kb
  proof
    let t be Point of TOP-REAL 2;
    assume that
A5: t in K0 and
A6: t<>0.TOP-REAL 2;
A7: ex p3 being Point of TOP-REAL 2 st p3=t & -1<p3`1 & p3 `1<1 & -1<p3`2
    & p3`2<1 by A1,A5;
    now
      assume
A8:   Out_In_Sq.t in K0 \/ Kb;
      now
        per cases by A8,XBOOLE_0:def 3;
        case
          Out_In_Sq.t in K0;
          then consider p4 being Point of TOP-REAL 2 such that
A9:       p4=Out_In_Sq.t and
A10:      -1<p4`1 and
A11:      p4`1<1 and
A12:      -1<p4`2 and
A13:      p4`2<1 by A1;
          now
            per cases;
            case
A14:          t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1;
              then Out_In_Sq.t=|[1/t`1,t`2/t`1/t`1]| by A6,Def1;
              then
A15:          p4`1=1/t`1 by A9,EUCLID:52;
              now
                per cases;
                case
A16:              t`1>=0;
                  now
                    per cases by A16;
                    case
A17:                  t`1>0;
                      then 1/t`1*t`1<1 *t`1 by A11,A15,XREAL_1:68;
                      hence contradiction by A7,A17,XCMPLX_1:87;
                    end;
                    case
A18:                  t`1=0;
                      then t`2=0 by A14;
                      hence contradiction by A6,A18,EUCLID:53,54;
                    end;
                  end;
                  hence contradiction;
                end;
                case
A19:              t`1<0;
                  then (-1)*t`1>1/t`1*t`1 by A10,A15,XREAL_1:69;
                  then (-1)*t`1>1 by A19,XCMPLX_1:87;
                  then --t`1<=-1 by XREAL_1:24;
                  hence contradiction by A7;
                end;
              end;
              hence contradiction;
            end;
            case
A20:          not(t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1);
              then Out_In_Sq.t=|[t`1/t`2/t`2,1/t`2]| by A6,Def1;
              then
A21:          p4`2=1/t`2 by A9,EUCLID:52;
              now
                per cases;
                case
A22:              t`2>=0;
                  now
                    per cases by A22;
                    case
A23:                  t`2>0;
                      then 1/t`2*t`2<1 *t`2 by A13,A21,XREAL_1:68;
                      hence contradiction by A7,A23,XCMPLX_1:87;
                    end;
                    case
                      t`2=0;
                      hence contradiction by A20;
                    end;
                  end;
                  hence contradiction;
                end;
                case
A24:              t`2<0;
                  then (-1)*t`2>1/t`2*t`2 by A12,A21,XREAL_1:69;
                  then (-1)*t`2>1 by A24,XCMPLX_1:87;
                  then --t`2<=-1 by XREAL_1:24;
                  hence contradiction by A7;
                end;
              end;
              hence contradiction;
            end;
          end;
          hence contradiction;
        end;
        case
          Out_In_Sq.t in Kb;
          then consider p4 being Point of TOP-REAL 2 such that
A25:      p4=Out_In_Sq.t and
A26:      -1=p4`1 & -1<=p4`2 & p4`2<=1 or p4`1=1 & -1<=p4`2 & p4`2<=
1 or -1 =p4`2 & -1<=p4`1 & p4`1<=1 or 1=p4`2 & -1<=p4`1 & p4`1<=1 by A1;
          now
            per cases;
            case
A27:          t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1;
              then
A28:          Out_In_Sq.t=|[1/t`1,t`2/t`1/t`1]| by A6,Def1;
              then
A29:          p4`1=1/t`1 by A25,EUCLID:52;
              now
                per cases by A26;
                case
                  -1=p4`1 & -1<=p4`2 & p4`2<=1;
                  then
A30:              (t`1)*(t`1)" =-t`1 by A29;
                  now
                    per cases;
                    case
                      t`1<>0;
                      then -t`1=1 by A30,XCMPLX_0:def 7;
                      hence contradiction by A7;
                    end;
                    case
A31:                  t`1=0;
                      then t`2=0 by A27;
                      hence contradiction by A6,A31,EUCLID:53,54;
                    end;
                  end;
                  hence contradiction;
                end;
                case
                  p4`1=1 & -1<=p4`2 & p4`2<=1;
                  then
A32:              (t`1)*(t`1)"=t`1 by A29;
                  now
                    per cases;
                    case
                      t`1<>0;
                      hence contradiction by A7,A32,XCMPLX_0:def 7;
                    end;
                    case
A33:                  t`1=0;
                      then t`2=0 by A27;
                      hence contradiction by A6,A33,EUCLID:53,54;
                    end;
                  end;
                  hence contradiction;
                end;
                case
A34:              -1=p4`2 & -1<=p4`1 & p4`1<=1;
                  reconsider K01=K0a as non empty Subset of (TOP-REAL 2)|D by
A2,Th17;
A35:              the carrier of ((TOP-REAL 2)|D)|K01=[#](((TOP-REAL 2)|
                  D)|K01)
                    .=K01 by PRE_TOPC:def 5;
A36:              dom (Out_In_Sq|K01)=(dom Out_In_Sq) /\ K01 by RELAT_1:61
                    .=D /\ K01 by A3,FUNCT_2:def 1
                    .=[#]((TOP-REAL 2)|D) /\ K01 by PRE_TOPC:def 5
                    .=(the carrier of ((TOP-REAL 2)|D)) /\ K01
                    .=K01 by XBOOLE_1:28;
                  t in K01 by A6,A27;
                  then
A37:              (Out_In_Sq|K01).t in rng (Out_In_Sq|K01) by A36,FUNCT_1:3;
                  rng (Out_In_Sq|K01) c= the carrier of ((TOP-REAL 2)|D)
                  |K01 by Th15;
                  then
A38:              (Out_In_Sq|K01).t in the carrier of (( TOP-REAL 2)|D)|
                  K01 by A37;
                  t in K01 by A6,A27;
                  then Out_In_Sq.t in K0a by A38,A35,FUNCT_1:49;
                  then
A39:              ex p5 being Point of TOP-REAL 2 st p5=p4 &( p5`2<=p5`1
& -p5`1<=p5`2 or p5`2>=p5`1 & p5`2<=-p5`1)& p5<>0.TOP-REAL 2 by A25;
                  now
                    per cases by A34,A39,XREAL_1:24;
                    case
A40:                  p4`1>=1;
                      then t`2/t`1/t`1=(t`2/t`1)*1 by A29,A34,XXREAL_0:1
                        .=t`2*1 by A29,A34,A40,XXREAL_0:1
                        .=t`2;
                      hence contradiction by A7,A25,A28,A34,EUCLID:52;
                    end;
                    case
A41:                  -1>=p4`1;
                      then t`2/t`1/t`1=(t`2/t`1)*(-1) by A29,A34,XXREAL_0:1
                        .=-(t`2/t`1)
                        .=-(t`2*(-1)) by A29,A34,A41,XXREAL_0:1
                        .=t`2;
                      hence contradiction by A7,A25,A28,A34,EUCLID:52;
                    end;
                  end;
                  hence contradiction;
                end;
                case
A42:              1=p4`2 & -1<=p4`1 & p4`1<=1;
                  reconsider K01=K0a as non empty Subset of (TOP-REAL 2)|D by
A2,Th17;
                  t in K01 by A6,A27;
                  then
A43:              Out_In_Sq.t=(Out_In_Sq|K01).t by FUNCT_1:49;
                  dom (Out_In_Sq|K01)=(dom Out_In_Sq) /\ K01 by RELAT_1:61
                    .=D /\ K01 by A3,FUNCT_2:def 1
                    .=[#]((TOP-REAL 2)|D) /\ K01 by PRE_TOPC:def 5
                    .=(the carrier of ((TOP-REAL 2)|D)) /\ K01
                    .=K01 by XBOOLE_1:28;
                  then t in dom (Out_In_Sq|K01) by A6,A27;
                  then
A44:              (Out_In_Sq|K01).t in rng (Out_In_Sq|K01) by FUNCT_1:3;
                  rng (Out_In_Sq|K01) c= the carrier of ((TOP-REAL 2)|D)
                  |K01 by Th15;
                  then
A45:              (Out_In_Sq|K01).t in the carrier of (( TOP-REAL 2)|D)|
                  K01 by A44;
                  the carrier of ((TOP-REAL 2)|D)|K01=[#](((TOP-REAL 2)|
                  D)|K01)
                    .=K01 by PRE_TOPC:def 5;
                  then
A46:              ex p5 being Point of TOP-REAL 2 st p5=p4 &( p5`2<=p5`1
& -p5`1<=p5`2 or p5`2>=p5`1 & p5`2<=-p5`1)& p5<>0.TOP-REAL 2 by A25,A45,A43;
                  now
                    per cases by A42,A46,XREAL_1:25;
                    case
A47:                  p4`1>=1;
                      then t`2/t`1/t`1=(t`2/t`1)*1 by A29,A42,XXREAL_0:1
                        .=t`2*1 by A29,A42,A47,XXREAL_0:1
                        .=t`2;
                      hence contradiction by A7,A25,A28,A42,EUCLID:52;
                    end;
                    case
A48:                  -1>=p4`1;
                      then t`2/t`1/t`1=(t`2/t`1)*(-1) by A29,A42,XXREAL_0:1
                        .=-(t`2/t`1)
                        .=-(t`2*(-1)) by A29,A42,A48,XXREAL_0:1
                        .=t`2;
                      hence contradiction by A7,A25,A28,A42,EUCLID:52;
                    end;
                  end;
                  hence contradiction;
                end;
              end;
              hence contradiction;
            end;
            case
A49:          not(t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1);
              then
A50:          Out_In_Sq.t=|[t`1/t`2/t`2,1/t`2]| by A6,Def1;
              then
A51:          p4`2=1/t`2 by A25,EUCLID:52;
              now
                per cases by A26;
                case
                  -1=p4`2 & -1<=p4`1 & p4`1<=1;
                  then
A52:              (t`2)*(t`2)"=-t`2 by A51;
                  now
                    per cases;
                    case
                      t`2<>0;
                      then -t`2=1 by A52,XCMPLX_0:def 7;
                      hence contradiction by A7;
                    end;
                    case
                      t`2=0;
                      hence contradiction by A49;
                    end;
                  end;
                  hence contradiction;
                end;
                case
                  p4`2=1 & -1<=p4`1 & p4`1<=1;
                  then
A53:              (t`2)*(t`2)" =t`2 by A51;
                  now
                    per cases;
                    case
                      t`2<>0;
                      hence contradiction by A7,A53,XCMPLX_0:def 7;
                    end;
                    case
                      t`2=0;
                      hence contradiction by A49;
                    end;
                  end;
                  hence contradiction;
                end;
                case
A54:              -1=p4`1 & -1<=p4`2 & p4`2<=1;
                  reconsider K11=K1a as non empty Subset of (TOP-REAL 2)|D by
A2,Th18;
A55:              dom (Out_In_Sq|K11)=(dom Out_In_Sq) /\ K11 by RELAT_1:61
                    .=D /\ K11 by A3,FUNCT_2:def 1
                    .=[#]((TOP-REAL 2)|D) /\ K11 by PRE_TOPC:def 5
                    .=(the carrier of ((TOP-REAL 2)|D)) /\ K11
                    .=K11 by XBOOLE_1:28;
A56:              t`1<=t`2 & -t`2<=t`1 or t`1>=t`2 & t`1<=-t`2 by A49,Th13;
                  then t in K11 by A6;
                  then
A57:              Out_In_Sq.t=(Out_In_Sq|K11).t by FUNCT_1:49;
                  t in K11 by A6,A56;
                  then
A58:              (Out_In_Sq|K11).t in rng (Out_In_Sq|K11) by A55,FUNCT_1:3;
                  rng (Out_In_Sq|K11) c= the carrier of ((TOP-REAL 2)|D)
                  |K11 by Th16;
                  then
A59:              (Out_In_Sq|K11).t in the carrier of (( TOP-REAL 2)|D)|
                  K11 by A58;
                  the carrier of ((TOP-REAL 2)|D)|K11=[#](((TOP-REAL 2)|
                  D)|K11)
                    .=K11 by PRE_TOPC:def 5;
                  then
A60:              ex p5 being Point of TOP-REAL 2 st p5=p4 &( p5`1<=p5`2
& -p5`2<=p5`1 or p5`1>=p5`2 & p5`1<=-p5`2)& p5<>0.TOP-REAL 2 by A25,A59,A57;
                  now
                    per cases by A54,A60,XREAL_1:24;
                    case
A61:                  p4`2>=1;
                      then t`1/t`2/t`2=(t`1/t`2)*1 by A51,A54,XXREAL_0:1
                        .=t`1*1 by A51,A54,A61,XXREAL_0:1
                        .=t`1;
                      hence contradiction by A7,A25,A50,A54,EUCLID:52;
                    end;
                    case
A62:                  -1>=p4`2;
                      then t`1/t`2/t`2=(t`1/t`2)*(-1) by A51,A54,XXREAL_0:1
                        .=-(t`1/t`2)
                        .=-(t`1*(-1)) by A51,A54,A62,XXREAL_0:1
                        .=t`1;
                      hence contradiction by A7,A25,A50,A54,EUCLID:52;
                    end;
                  end;
                  hence contradiction;
                end;
                case
A63:              1=p4`1 & -1<=p4`2 & p4`2<=1;
                  reconsider K11=K1a as non empty Subset of (TOP-REAL 2)|D by
A2,Th18;
A64:              the carrier of ((TOP-REAL 2)|D)|K11=[#](((TOP-REAL 2)|
                  D)|K11)
                    .=K11 by PRE_TOPC:def 5;
A65:              dom (Out_In_Sq|K11)=(dom Out_In_Sq) /\ K11 by RELAT_1:61
                    .=D /\ K11 by A3,FUNCT_2:def 1
                    .=[#]((TOP-REAL 2)|D) /\ K11 by PRE_TOPC:def 5
                    .=(the carrier of ((TOP-REAL 2)|D)) /\ K11
                    .=K11 by XBOOLE_1:28;
A66:              t`1<=t`2 & -t`2<=t`1 or t`1>=t`2 & t`1<=-t`2 by A49,Th13;
                  then t in K11 by A6;
                  then
A67:              (Out_In_Sq|K11).t in rng (Out_In_Sq|K11) by A65,FUNCT_1:3;
                  rng (Out_In_Sq|K11) c= the carrier of ((TOP-REAL 2)|D)
                  |K11 by Th16;
                  then
A68:              (Out_In_Sq|K11).t in the carrier of (( TOP-REAL 2)|D)|
                  K11 by A67;
                  t in K11 by A6,A66;
                  then Out_In_Sq.t in K1a by A68,A64,FUNCT_1:49;
                  then
A69:              ex p5 being Point of TOP-REAL 2 st p5=p4 &( p5`1<=p5`2
& -p5`2<=p5`1 or p5`1>=p5`2 & p5`1<=-p5`2)& p5<>0.TOP-REAL 2 by A25;
                  now
                    per cases by A63,A69,XREAL_1:25;
                    case
A70:                  p4`2>=1;
                      then t`1/t`2/t`2=(t`1/t`2)*1 by A51,A63,XXREAL_0:1
                        .=t`1*1 by A51,A63,A70,XXREAL_0:1
                        .=t`1;
                      hence contradiction by A7,A25,A50,A63,EUCLID:52;
                    end;
                    case
A71:                  -1>=p4`2;
                      then t`1/t`2/t`2=(t`1/t`2)*(-1) by A51,A63,XXREAL_0:1
                        .=-(t`1/t`2)
                        .=-(t`1*(-1)) by A51,A63,A71,XXREAL_0:1
                        .=t`1;
                      hence contradiction by A7,A25,A50,A63,EUCLID:52;
                    end;
                  end;
                  hence contradiction;
                end;
              end;
              hence contradiction;
            end;
          end;
          hence contradiction;
        end;
      end;
      hence contradiction;
    end;
    hence thesis;
  end;
A72: for t being Point of TOP-REAL 2 st not t in K0 \/ Kb holds Out_In_Sq.t
  in K0
  proof
    let t be Point of TOP-REAL 2;
    assume
A73: not t in K0 \/ Kb;
    then
A74: not t in K0 by XBOOLE_0:def 3;
    then
A75: not t=0.TOP-REAL 2 by A1,Th3;
    then not t in {0.TOP-REAL 2} by TARSKI:def 1;
    then t in NonZero TOP-REAL 2 by XBOOLE_0:def 5;
    then Out_In_Sq.t in NonZero TOP-REAL 2 by FUNCT_2:5;
    then reconsider p4=Out_In_Sq.t as Point of TOP-REAL 2;
A76: not t in Kb by A73,XBOOLE_0:def 3;
    now
      per cases;
      case
A77:    t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1;
A78:    now
          per cases;
          case
A79:        t`1>0;
            now
              per cases;
              case
A80:            t`2>0;
                -1>=t`1 or t`1>=1 or -1>=t`2 or t`2>=1 by A1,A74;
                then
A81:            t`1>=1 by A77,A79,A80,XXREAL_0:2;
                not t`1=1 by A1,A76,A77;
                then
A82:            t`1>1 by A81,XXREAL_0:1;
                then t`1<(t`1)^2 by SQUARE_1:14;
                then (t`2)<(t`1)^2 by A77,A79,XXREAL_0:2;
                then t`2/t`1<(t`1)^2/t`1 by A79,XREAL_1:74;
                then t`2/t`1<(t`1) by A79,XCMPLX_1:89;
                then
A83:            t`2/t`1/t`1<(t`1)/t`1 by A79,XREAL_1:74;
                0<t`2/t`1 by A79,A80,XREAL_1:139;
                then
A84:            (-1)*t`1/t`1< t`2/t`1/t`1 by A79,XREAL_1:74;
                t`1/t`1>1/t`1 by A82,XREAL_1:74;
                hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1 by
A79,A84,A83,XCMPLX_1:60,89;
              end;
              case
A85:            t`2<=0;
A86:            now
                  assume t`1<1;
                  then -1>=t`2 by A1,A74,A79,A85;
                  then -t`1<=-1 by A77,A79,XXREAL_0:2;
                  hence t`1>=1 by XREAL_1:24;
                end;
                not t`1=1 by A1,A76,A77;
                then
A87:            t`1>1 by A86,XXREAL_0:1;
                then
A88:            t`1<(t`1)^2 by SQUARE_1:14;
                --t`1>=-t`2 by A77,A79,XREAL_1:24;
                then (t`1)^2 >-t`2 by A88,XXREAL_0:2;
                then (t`1)^2/t`1 >(-t`2)/t`1 by A79,XREAL_1:74;
                then t`1>-(t`2/t`1) by A79,XCMPLX_1:89;
                then -t`1<--(t`2/t`1) by XREAL_1:24;
                then
A89:            (-1)*t`1/t`1< t`2/t`1/t`1 by A79,XREAL_1:74;
                t`1/t`1>1/t`1 by A87,XREAL_1:74;
                hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1 by
A79,A85,A89,XCMPLX_1:60,89;
              end;
            end;
            hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1;
          end;
          case
A90:        t`1<=0;
            now
              per cases by A90;
              case
A91:            t`1=0;
                then t`2=0 by A77;
                hence contradiction by A1,A74,A91;
              end;
              case
A92:            t`1<0;
                now
                  per cases;
                  case
A93:                t`2>0;
                    -1>=t`1 or t`1>=1 or -1>=t`2 or t`2>=1 by A1,A74;
                    then t`1<=-1 or 1<=-t`1 by A77,A92,XXREAL_0:2;
                    then
A94:                t`1<=-1 or -1>=--t`1 by XREAL_1:24;
                    not t`1=-1 by A1,A76,A77;
                    then
A95:                t`1<-1 by A94,XXREAL_0:1;
                    then t`1/t`1>(-1)/t`1 by XREAL_1:75;
                    then
A96:                -(t`1/t`1)<-((-1)/t`1) by XREAL_1:24;
                    -t`1<(t`1)^2 by A95,SQUARE_1:46;
                    then (t`2)<(t`1)^2 by A77,A92,XXREAL_0:2;
                    then t`2/t`1>(t`1)^2/t`1 by A92,XREAL_1:75;
                    then t`2/t`1>(t`1) by A92,XCMPLX_1:89;
                    then
A97:                t`2/t`1/t`1<(t`1)/t`1 by A92,XREAL_1:75;
                    0>t`2/t`1 by A92,A93,XREAL_1:142;
                    then (-1)*t`1/t`1< t`2/t`1/t`1 by A92,XREAL_1:75;
                    hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1
                    by A92,A96,A97,XCMPLX_1:60;
                  end;
                  case
A98:                t`2<=0;
                    then -1>=t`1 or -1>=t`2 by A1,A74,A92;
                    then
A99:                t`1<=-1 by A77,A92,XXREAL_0:2;
                    not t`1=-1 by A1,A76,A77;
                    then
A100:               t`1< -1 by A99,XXREAL_0:1;
                    then
A101:               -t`1<(t`1)^2 by SQUARE_1:46;
                    t`1<=t`2 by A77,A92;
                    then -t`1>=-t`2 by XREAL_1:24;
                    then (t`1)^2 >-t`2 by A101,XXREAL_0:2;
                    then (t`1)^2/t`1 <(-t`2)/t`1 by A92,XREAL_1:75;
                    then t`1<-(t`2/t`1) by A92,XCMPLX_1:89;
                    then -t`1>--(t`2/t`1) by XREAL_1:24;
                    then
A102:               (-1)*t`1/t`1< t`2/t`1/t`1 by A92,XREAL_1:75;
                    t`1/t`1> (-1)/t`1 by A100,XREAL_1:75;
                    then 1> (-1)/t`1 by A92,XCMPLX_1:60;
                    then -1<-(-1)/t`1 by XREAL_1:24;
                    hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1
                    by A92,A98,A102,XCMPLX_1:89;
                  end;
                end;
                hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1;
              end;
            end;
            hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1;
          end;
        end;
        Out_In_Sq.t=|[1/t`1,t`2/t`1/t`1]| by A75,A77,Def1;
        then p4`1=1/t`1 & p4`2=t`2/t`1/t`1 by EUCLID:52;
        hence thesis by A1,A78;
      end;
      case
A103:   not(t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1);
        then
A104:   t`1<=t`2 & -t`2<=t`1 or t`1>=t`2 & t`1<=-t`2 by Th13;
A105:   now
          per cases;
          case
A106:       t`2>0;
            now
              per cases;
              case
A107:           t`1>0;
A108:           -1>=t`2 or t`2>=1 or -1>=t`1 or t`1>=1 by A1,A74;
                not t`2=1 by A1,A76,A103,A107;
                then
A109:           t`2>1 by A103,A106,A107,A108,XXREAL_0:1,2;
                then t`2<(t`2)^2 by SQUARE_1:14;
                then (t`1)<(t`2)^2 by A103,A106,XXREAL_0:2;
                then t`1/t`2<(t`2)^2/t`2 by A106,XREAL_1:74;
                then t`1/t`2<(t`2) by A106,XCMPLX_1:89;
                then
A110:           t`1/t`2/t`2<(t`2)/t`2 by A106,XREAL_1:74;
                0<t`1/t`2 by A106,A107,XREAL_1:139;
                then
A111:           (-1)*t`2/t`2< t`1/t`2/t`2 by A106,XREAL_1:74;
                t`2/t`2>1/t`2 by A109,XREAL_1:74;
                hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1 by
A106,A111,A110,XCMPLX_1:60,89;
              end;
              case
A112:           t`1<=0;
A113:           now
                  assume t`2<1;
                  then -1>=t`1 by A1,A74,A106,A112;
                  then -t`2<=-1 by A104,A106,XXREAL_0:2;
                  hence t`2>=1 by XREAL_1:24;
                end;
                not t`2=1 by A1,A76,A104;
                then
A114:           t`2>1 by A113,XXREAL_0:1;
                then t`2<(t`2)^2 by SQUARE_1:14;
                then (t`2)^2 >-t`1 by A103,A106,XXREAL_0:2;
                then (t`2)^2/t`2 >(-t`1)/t`2 by A106,XREAL_1:74;
                then t`2>-(t`1/t`2) by A106,XCMPLX_1:89;
                then -t`2<--(t`1/t`2) by XREAL_1:24;
                then
A115:           (-1)*t`2/t`2< t`1/t`2/t`2 by A106,XREAL_1:74;
                t`2/t`2>1/t`2 by A114,XREAL_1:74;
                hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1 by
A106,A112,A115,XCMPLX_1:60,89;
              end;
            end;
            hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1;
          end;
          case
A116:       t`2<=0;
            then
A117:       t`2<0 by A103;
A118:       t`1<=t`2 or t`1<=-t`2 by A103,Th13;
            now
              per cases;
              case
A119:           t`1>0;
                -1>=t`2 or t`2>=1 or -1>=t`1 or t`1>=1 by A1,A74;
                then t`2<=-1 or 1<=-t`2 by A104,A116,XXREAL_0:2;
                then
A120:           t`2<=-1 or -1>=--t`2 by XREAL_1:24;
                not t`2=-1 by A1,A76,A104;
                then
A121:           t`2<-1 by A120,XXREAL_0:1;
                then t`2/t`2>(-1)/t`2 by XREAL_1:75;
                then
A122:           -(t`2/t`2)<-((-1)/t`2) by XREAL_1:24;
                -t`2<(t`2)^2 by A121,SQUARE_1:46;
                then (t`1)<(t`2)^2 by A116,A118,XXREAL_0:2;
                then t`1/t`2>(t`2)^2/t`2 by A117,XREAL_1:75;
                then t`1/t`2>(t`2) by A117,XCMPLX_1:89;
                then
A123:           t`1/t`2/t`2<(t`2)/t`2 by A117,XREAL_1:75;
                0>t`1/t`2 by A117,A119,XREAL_1:142;
                then (-1)*t`2/t`2< t`1/t`2/t`2 by A117,XREAL_1:75;
                hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1 by
A117,A122,A123,XCMPLX_1:60;
              end;
              case
A124:           t`1<=0;
A125:           not t`2=-1 by A1,A76,A104;
                -1>=t`2 or -1>=t`1 by A1,A74,A116,A124;
                then
A126:           t`2< -1 by A103,A116,A125,XXREAL_0:1,2;
                then
A127:           -t`2<(t`2)^2 by SQUARE_1:46;
                -t`2>=-t`1 by A103,A116,XREAL_1:24;
                then (t`2)^2 >-t`1 by A127,XXREAL_0:2;
                then (t`2)^2/t`2 <(-t`1)/t`2 by A117,XREAL_1:75;
                then t`2<-(t`1/t`2) by A117,XCMPLX_1:89;
                then -t`2>--(t`1/t`2) by XREAL_1:24;
                then
A128:           (-1)*t`2/t`2< t`1/t`2/t`2 by A117,XREAL_1:75;
                t`2/t`2> (-1)/t`2 by A126,XREAL_1:75;
                then 1> (-1)/t`2 by A117,XCMPLX_1:60;
                then -1<-(-1)/t`2 by XREAL_1:24;
                hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1 by
A103,A116,A124,A128,XCMPLX_1:89;
              end;
            end;
            hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1;
          end;
        end;
        Out_In_Sq.t=|[t`1/t`2/t`2,1/t`2]| by A75,A103,Def1;
        then p4`2=1/t`2 & p4`1=t`1/t`2/t`2 by EUCLID:52;
        hence thesis by A1,A105;
      end;
    end;
    hence thesis;
  end;
A129: D =(NonZero TOP-REAL 2) by A1,SUBSET_1:def 4;
  for x1,x2 being object st x1 in dom Out_In_Sq & x2 in dom Out_In_Sq &
  Out_In_Sq.x1=Out_In_Sq.x2 holds x1=x2
  proof
A130: K1a c= D
    proof
      let x be object;
      assume x in K1a;
      then
A131: ex p8 being Point of TOP-REAL 2 st x=p8 &( p8`1<=p8`2 & - p8`2<=p8`1
      or p8`1>=p8`2 & p8`1<=-p8`2)& p8<>0.TOP-REAL 2;
      then not x in {0.TOP-REAL 2} by TARSKI:def 1;
      hence thesis by A3,A131,XBOOLE_0:def 5;
    end;
A132: (1.REAL 2)<>0.TOP-REAL 2 by Lm1,REVROT_1:19;
    (1.REAL 2)`1<=(1.REAL 2)`2 & -(1.REAL 2)`2<=(1.REAL 2)`1 or (1.REAL 2
    ) `1>=(1.REAL 2)`2 & (1.REAL 2)`1<=-(1.REAL 2)`2 by Th5;
    then
A133: 1.REAL 2 in K1a by A132;
    the carrier of (TOP-REAL 2)|D=[#]((TOP-REAL 2)|D) .=D by PRE_TOPC:def 5;
    then reconsider K11=K1a as non empty Subset of ((TOP-REAL 2)|D) by A133
,A130;
    reconsider K01=K0a as non empty Subset of ((TOP-REAL 2)|D) by A2,Th17;
    let x1,x2 be object;
    assume that
A134: x1 in dom Out_In_Sq and
A135: x2 in dom Out_In_Sq and
A136: Out_In_Sq.x1=Out_In_Sq.x2;
    NonZero TOP-REAL 2<>{} by Th9;
    then
A137: dom Out_In_Sq=NonZero TOP-REAL 2 by FUNCT_2:def 1;
    then
A138: x2 in D by A1,A135,SUBSET_1:def 4;
    reconsider p1=x1,p2=x2 as Point of TOP-REAL 2 by A134,A135,XBOOLE_0:def 5;
A139: D c= K01 \/ K11
    proof
      let x be object;
      assume
A140: x in D;
      then reconsider px=x as Point of TOP-REAL 2;
      not x in {0.TOP-REAL 2} by A129,A140,XBOOLE_0:def 5;
      then (px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1) & px<>0.
TOP-REAL 2 or (px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2) & px<>0.
      TOP-REAL 2 by TARSKI:def 1,XREAL_1:25;
      then x in K01 or x in K11;
      hence thesis by XBOOLE_0:def 3;
    end;
A141: x1 in D by A1,A134,A137,SUBSET_1:def 4;
    now
      per cases by A139,A141,XBOOLE_0:def 3;
      case
        x1 in K01;
        then
A142:   ex p7 being Point of TOP-REAL 2 st p1=p7 &( p7`2<=p7`1 & -p7`1<=p7
        `2 or p7`2>=p7`1 & p7`2<=-p7`1)& p7<>0.TOP-REAL 2;
        then
A143:   Out_In_Sq.p1=|[1/p1`1,p1`2/p1`1/p1`1]| by Def1;
        now
          per cases by A139,A138,XBOOLE_0:def 3;
          case
            x2 in K0a;
            then
            ex p8 being Point of (TOP-REAL 2) st p2=p8 &( p8`2<=p8`1 & -p8
            `1<=p8`2 or p8`2>=p8`1 & p8`2<=-p8`1)& p8<>0.TOP-REAL 2;
            then
A144:       |[1/p2`1,p2`2/p2`1/p2`1]| =|[1/p1`1,p1`2/p1`1/p1`1]| by A136,A143
,Def1;
A145:       p1=|[p1`1,p1`2]| by EUCLID:53;
            set qq=|[1/p2`1,p2`2/p2`1/p2`1]|;
A146:       (1/p1`1)"=(p1`1)"" .=p1`1;
A147:       now
              assume
A148:         p1`1=0;
              then p1`2=0 by A142;
              hence contradiction by A142,A148,EUCLID:53,54;
            end;
            qq`1=1/p2`1 by EUCLID:52;
            then
A149:       1/p1`1= 1/p2`1 by A144,EUCLID:52;
            qq`2=p2`2/p2`1/p2`1 by EUCLID:52;
            then p1`2/p1`1= p2`2/p1`1 by A144,A149,A146,A147,EUCLID:52
,XCMPLX_1:53;
            then p1`2=p2`2 by A147,XCMPLX_1:53;
            hence thesis by A149,A146,A145,EUCLID:53;
          end;
          case
A150:       x2 in K1a & not x2 in K0a;
A151:       now
              assume
A152:         p1`1=0;
              then p1`2=0 by A142;
              hence contradiction by A142,A152,EUCLID:53,54;
            end;
A153:       now
              per cases by A142;
              case
A154:           p1`2<=p1`1 & -p1`1<=p1`2;
                then -p1`1 <= p1`1 by XXREAL_0:2;
                then p1`1>=0;
                then p1`2/p1`1<=p1`1/p1`1 by A154,XREAL_1:72;
                hence p1`2/p1`1<=1 by A151,XCMPLX_1:60;
              end;
              case
A155:           p1`2>=p1`1 & p1`2<=-p1`1;
                then -p1`1 >= p1`1 by XXREAL_0:2;
                then p1`1 <= 0;
                then p1`2/p1`1<=p1`1/p1`1 by A155,XREAL_1:73;
                hence p1`2/p1`1<=1 by A151,XCMPLX_1:60;
              end;
            end;
A156:       now
              per cases by A142;
              case
A157:           p1`2<=p1`1 & -p1`1<=p1`2;
                then -p1`1 <= p1`1 by XXREAL_0:2;
                then p1`1>=0;
                then (-p1`1)/p1`1<=p1`2/p1`1 by A157,XREAL_1:72;
                hence -1<=p1`2/p1`1 by A151,XCMPLX_1:197;
              end;
              case
A158:              p1`2>=p1`1 & p1`2<=-p1`1;
                -p1`2>=--p1`1 & p1`1 <= 0 by XREAL_1:24,A158;
                then (-p1`2)/(-p1`1)>=p1`1/(-p1`1) by XREAL_1:72;
                then (-p1`2)/(-p1`1)>= -1 by A151,XCMPLX_1:198;
                hence -1<=p1`2/p1`1 by XCMPLX_1:191;
              end;
            end;
A159:       ex p8 being Point of (TOP-REAL 2) st p2=p8 &( p8`1<=p8`2 & -p8
            `2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2)& p8<>0.TOP-REAL 2 by A150;
A160:       now
              assume
A161:         p2`2=0;
              then p2`1=0 by A159;
              hence contradiction by A159,A161,EUCLID:53,54;
            end;
            not((p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 & p2`2<=-p2`1) &
            p2 <> 0.TOP-REAL 2) by A150;
            then
A162:       Out_In_Sq.p2=|[p2`1/p2`2/p2`2,1/p2`2]| by A159,Def1;
            then p1`2/p1`1/p1`1=1/p2`2 by A136,A143,SPPOL_2:1;
            then
A163:       p1`2/p1`1=1/p2`2*p1`1 by A151,XCMPLX_1:87
              .= p1`1/p2`2;
            1/p1`1=p2`1/p2`2/p2`2 by A136,A143,A162,SPPOL_2:1;
            then
A164:       p2`1/p2`2=1/p1`1*p2`2 by A160,XCMPLX_1:87
              .= p2`2/p1`1;
            then
A165:       (p2`1/p2`2)* (p1`2/p1`1)=1 by A160,A151,A163,XCMPLX_1:112;
A166:       (p2`1/p2`2)* (p1`2/p1`1)*p1`1=1 *p1`1 by A160,A151,A164,A163,
XCMPLX_1:112;
            then
A167:       p1`2<>0 by A151;
A168:       ex p9 being Point of (TOP-REAL 2) st p2=p9 &( p9`1<=p9`2 & -p9
            `2<=p9`1 or p9`1>=p9`2 & p9`1<=-p9`2)& p9<>0.TOP-REAL 2 by A150;
A169:       now
              per cases by A168;
              case
A170:           p2`1<=p2`2 & -p2`2<=p2`1;
                then -p2`2 <= p2`2 by XXREAL_0:2;
                then p2`2>=0;
                then (-p2`2)/p2`2<=p2`1/p2`2 by A170,XREAL_1:72;
                hence -1<=p2`1/p2`2 by A160,XCMPLX_1:197;
              end;
              case
A171:              p2`1>=p2`2 & p2`1<=-p2`2;
                -p2`1>=--p2`2 & p2`2 <= 0 by XREAL_1:24,A171;
                then (-p2`1)/(-p2`2)>=p2`2/(-p2`2) by XREAL_1:72;
                then (-p2`1)/(-p2`2)>= -1 by A160,XCMPLX_1:198;
                hence -1<=p2`1/p2`2 by XCMPLX_1:191;
              end;
            end;
            (p2`1/p2`2)* ((p1`2/p1`1)*p1`1)=p1`1 by A166;
            then
A172:       (p2`1/p2`2)*p1`2=p1`1 by A151,XCMPLX_1:87;
            then
A173:       (p2`1/p2`2)=p1`1/p1`2 by A167,XCMPLX_1:89;
A174:       now
              per cases by A168;
              case
A175:           p2`1<=p2`2 & -p2`2<=p2`1;
                then -p2`2 <= p2`2 by XXREAL_0:2;
                then p2`2>=0;
                then p2`1/p2`2<=p2`2/p2`2 by A175,XREAL_1:72;
                hence p2`1/p2`2<=1 by A160,XCMPLX_1:60;
              end;
              case
A176:           p2`1>=p2`2 & p2`1<=-p2`2;
                then -p2`2 >= p2`2 by XXREAL_0:2;
                then p2`2 <= 0;
                then p2`1/p2`2<=p2`2/p2`2 by A176,XREAL_1:73;
                hence p2`1/p2`2<=1 by A160,XCMPLX_1:60;
              end;
            end;
            now
              per cases;
              case
                0<=p2`1/p2`2;
                then
A177:           p1`2>0 & p1`1>=0 or p1`2<0 & p1`1<=0 by A151,A172;
                now
                  assume p1`2/p1`1<>1;
                  then p1`2/p1`1<1 by A153,XXREAL_0:1;
                  hence contradiction by A165,A174,A177,XREAL_1:162;
                end;
                then p1`2=(1)*p1`1 by A151,XCMPLX_1:87;
                then (p2`1/p2`2)*p2`2=(1)*p2`2 by A151,A173,XCMPLX_1:60
                  .=p2`2;
                then p2`1=p2`2 by A160,XCMPLX_1:87;
                hence contradiction by A150,A168;
              end;
              case
                0>p2`1/p2`2;
                then
A178:           p1`2<0 & p1`1>0 or p1`2>0 & p1`1<0 by A173,XREAL_1:143;
                now
                  assume p1`2/p1`1<>-1;
                  then -1<p1`2/p1`1 by A156,XXREAL_0:1;
                  hence contradiction by A165,A169,A178,XREAL_1:166;
                end;
                then p1`2=(-1)*p1`1 by A151,XCMPLX_1:87
                  .= -p1`1;
                then -p1`2 =p1`1;
                then p2`1/p2`2=-1 by A167,A173,XCMPLX_1:197;
                then p2`1=(-1)*p2`2 by A160,XCMPLX_1:87;
                then -p2`1=p2`2;
                hence contradiction by A150,A168;
              end;
            end;
            hence contradiction;
          end;
        end;
        hence thesis;
      end;
      case
        x1 in K1a;
        then
A179:   ex p7 being Point of TOP-REAL 2 st p1=p7 &( p7`1<=p7`2 & -p7`2<=
        p7`1 or p7`1>=p7`2 & p7`1<=-p7`2)& p7<>0.TOP-REAL 2;
        then
A180:   Out_In_Sq.p1=|[p1`1/p1`2/p1`2,1/p1`2]| by Th14;
        now
          per cases by A139,A138,XBOOLE_0:def 3;
          case
            x2 in K1a;
            then ex p8 being Point of (TOP-REAL 2) st p2=p8 &( p8`1<=p8`2 & -
            p8`2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2)& p8<>0.TOP-REAL 2;
            then
A181:       |[p2`1/p2`2/p2`2,1/p2`2]| =|[p1`1/p1`2/p1`2,1/p1`2]| by A136,A180
,Th14;
A182:       p1=|[p1`1,p1`2]| by EUCLID:53;
            set qq=|[p2`1/p2`2/p2`2,1/p2`2]|;
A183:       (1/p1`2)"=(p1`2)"" .=p1`2;
A184:       now
              assume
A185:         p1`2=0;
              then p1`1=0 by A179;
              hence contradiction by A179,A185,EUCLID:53,54;
            end;
            qq`2=1/p2`2 by EUCLID:52;
            then
A186:       1/p1`2= 1/p2`2 by A181,EUCLID:52;
            qq`1=p2`1/p2`2/p2`2 by EUCLID:52;
            then p1`1/p1`2= p2`1/p1`2 by A181,A186,A183,A184,EUCLID:52
,XCMPLX_1:53;
            then p1`1=p2`1 by A184,XCMPLX_1:53;
            hence thesis by A186,A183,A182,EUCLID:53;
          end;
          case
A187:       x2 in K0a & not x2 in K1a;
A188:       now
              assume
A189:         p1`2=0;
              then p1`1=0 by A179;
              hence contradiction by A179,A189,EUCLID:53,54;
            end;
A190:       now
              per cases by A179;
              case
A191:           p1`1<=p1`2 & -p1`2<=p1`1;
                then -p1`2 <= p1`2 by XXREAL_0:2;
                then p1`2>=0;
                then p1`1/p1`2<=p1`2/p1`2 by A191,XREAL_1:72;
                hence p1`1/p1`2<=1 by A188,XCMPLX_1:60;
              end;
              case
A192:           p1`1>=p1`2 & p1`1<=-p1`2;
                then
              -p1`2 >= p1`2 by XXREAL_0:2;
                then p1`2 <= 0;
                then p1`1/p1`2<=p1`2/p1`2 by A192,XREAL_1:73;
                hence p1`1/p1`2<=1 by A188,XCMPLX_1:60;
              end;
            end;
A193:       now
              per cases by A179;
              case
A194:           p1`1<=p1`2 & -p1`2<=p1`1;
                then -p1`2 <= p1`2 by XXREAL_0:2;
                then p1`2>=0;
                then (-p1`2)/p1`2<=p1`1/p1`2 by A194,XREAL_1:72;
                hence -1<=p1`1/p1`2 by A188,XCMPLX_1:197;
              end;
              case
A195:              p1`1>=p1`2 & p1`1<=-p1`2;
                -p1`1>=--p1`2 & p1`2 <= 0 by XREAL_1:24,A195;
                then (-p1`1)/(-p1`2)>=p1`2/(-p1`2) by XREAL_1:72;
                then (-p1`1)/(-p1`2)>= -1 by A188,XCMPLX_1:198;
                hence -1<=p1`1/p1`2 by XCMPLX_1:191;
              end;
            end;
A196:       ex p8 being Point of (TOP-REAL 2) st p2=p8 &( p8`2<=p8`1 & -
            p8`1<=p8`2 or p8`2>=p8`1 & p8`2<=-p8`1)& p8<>0.TOP-REAL 2 by A187;
A197:       now
              assume
A198:         p2`1=0;
              then p2`2=0 by A196;
              hence contradiction by A196,A198,EUCLID:53,54;
            end;
A199:       ex p9 being Point of (TOP-REAL 2) st p2=p9 &( p9`2<=p9`1 & -
            p9`1<=p9`2 or p9`2>=p9`1 & p9`2<=-p9`1)& p9<>0.TOP-REAL 2 by A187;
A200:       now
              per cases by A199;
              case
A201:           p2`2<=p2`1 & -p2`1<=p2`2;
                then -p2`1 <= p2`1 by XXREAL_0:2;
                then p2`1>=0;
                then (-p2`1)/p2`1<=p2`2/p2`1 by A201,XREAL_1:72;
                hence -1<=p2`2/p2`1 by A197,XCMPLX_1:197;
              end;
              case
A202:              p2`2>=p2`1 & p2`2<=-p2`1;
                -p2`2>=--p2`1 & p2`1 <= 0 by XREAL_1:24,A202;
                then (-p2`2)/(-p2`1)>=p2`1/(-p2`1) by XREAL_1:72;
                then (-p2`2)/(-p2`1)>= -1 by A197,XCMPLX_1:198;
                hence -1<=p2`2/p2`1 by XCMPLX_1:191;
              end;
            end;
A203:       Out_In_Sq.p2=|[1/p2`1,p2`2/p2`1/p2`1]| by A196,Def1;
            then 1/p1`2=p2`2/p2`1/p2`1 by A136,A180,SPPOL_2:1;
            then
A204:       p2`2/p2`1=1/p1`2*p2`1 by A197,XCMPLX_1:87
              .= p2`1/p1`2;
            p1`1/p1`2/p1`2=1/p2`1 by A136,A180,A203,SPPOL_2:1;
            then p1`1/p1`2=1/p2`1*p1`2 by A188,XCMPLX_1:87
              .= p1`2/p2`1;
            then
A205:       (p2`2/p2`1)* (p1`1/p1`2)=1 by A197,A188,A204,XCMPLX_1:112;
            then
A206:       p1`1<>0;
            (p2`2/p2`1)* (p1`1/p1`2)*p1`2=p1`2 by A205;
            then (p2`2/p2`1)* ((p1`1/p1`2)*p1`2)=p1`2;
            then (p2`2/p2`1)*p1`1=p1`2 by A188,XCMPLX_1:87;
            then
A207:       (p2`2/p2`1)=p1`2/p1`1 by A206,XCMPLX_1:89;
A208:       now
              per cases by A199;
              case
A209:           p2`2<=p2`1 & -p2`1<=p2`2;
                then -p2`1 <= p2`1 by XXREAL_0:2;
                then p2`1>=0;
                then p2`2/p2`1<=p2`1/p2`1 by A209,XREAL_1:72;
                hence p2`2/p2`1<=1 by A197,XCMPLX_1:60;
              end;
              case
A210:           p2`2>=p2`1 & p2`2<=-p2`1;
                then -p2`1 >= p2`1 by XXREAL_0:2;
                then p2`1 <= 0;
                then p2`2/p2`1<=p2`1/p2`1 by A210,XREAL_1:73;
                hence p2`2/p2`1<=1 by A197,XCMPLX_1:60;
              end;
            end;
            now
              per cases;
              case
                0<=p2`2/p2`1;
                then
A211:           p1`1>0 & p1`2>=0 or p1`1<0 & p1`2<=0 by A205,A206;
                now
                  assume p1`1/p1`2<>1;
                  then p1`1/p1`2<1 by A190,XXREAL_0:1;
                  hence contradiction by A205,A208,A211,XREAL_1:162;
                end;
                then p1`1=(1)*p1`2 by A188,XCMPLX_1:87;
                then (p2`2/p2`1)*p2`1 =(1)*p2`1 by A188,A207,XCMPLX_1:60
                  .=p2`1;
                then p2`2=p2`1 by A197,XCMPLX_1:87;
                hence contradiction by A187,A199;
              end;
              case
                0>p2`2/p2`1;
                then
A212:           p1`1<0 & p1`2>0 or p1`1>0 & p1`2<0 by A207,XREAL_1:143;
                now
                  assume p1`1/p1`2<>-1;
                  then -1<p1`1/p1`2 by A193,XXREAL_0:1;
                  hence contradiction by A205,A200,A212,XREAL_1:166;
                end;
                then p1`1=(-1)*p1`2 by A188,XCMPLX_1:87
                  .= -p1`2;
                then -p1`1 =p1`2;
                then p2`2/p2`1=-1 by A206,A207,XCMPLX_1:197;
                then p2`2=(-1)*p2`1 by A197,XCMPLX_1:87;
                then -p2`2=p2`1;
                hence contradiction by A187,A199;
              end;
            end;
            hence contradiction;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  then
A213: Out_In_Sq is one-to-one by FUNCT_1:def 4;
A214: for s being Point of TOP-REAL 2 st s in Kb holds Out_In_Sq.s=s
  proof
    let t be Point of TOP-REAL 2;
    assume t in Kb;
    then
A215: ex p4 being Point of TOP-REAL 2 st p4=t &( -1=p4`1 & -1<= p4`2 & p4`2
<=1 or p4`1=1 & -1<=p4`2 & p4`2<=1 or -1 =p4`2 & -1<=p4`1 & p4`1<=1 or 1=p4`2 &
    -1<=p4`1 & p4`1<=1) by A1;
    then
A216: t<>0.TOP-REAL 2 by EUCLID:52,54;
A217: not t=0.TOP-REAL 2 by A215,EUCLID:52,54;
    now
      per cases;
      case
A218:   t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1;
        then
A219:   Out_In_Sq.t=|[1/t`1,t`2/t`1/t`1]| by A217,Def1;
A220:   1<=t`1 & t`1>=-1 or 1>=t`1 & -1>=--t`1 by A215,A218,XREAL_1:24;
        now
          per cases by A215,A220,XXREAL_0:1;
          case
            t`1=1;
            hence thesis by A219,EUCLID:53;
          end;
          case
            t`1=-1;
            hence thesis by A219,EUCLID:53;
          end;
        end;
        hence thesis;
      end;
      case
A221:   not (t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1);
        then
A222:   Out_In_Sq.t=|[t`1/t`2/t`2,1/t`2]| by A216,Def1;
        now
          per cases by A215,A221;
          case
            t`2=1;
            hence thesis by A222,EUCLID:53;
          end;
          case
            t`2=-1;
            hence thesis by A222,EUCLID:53;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  ex h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D st h =Out_In_Sq & h
  is continuous by A2,Th40;
  hence thesis by A213,A4,A72,A214;
end;
