reserve p,p1,p2,p3,q,q1,q2 for Point of TOP-REAL 2,
  i for Nat,
  lambda for Real;

theorem Th72:
  for p1,p2,p3,p4 being Point of TOP-REAL 2,
  P being non empty compact Subset of TOP-REAL 2,
  f being Function of TOP-REAL 2,TOP-REAL 2 st P= circle(0,0,1) & f=Sq_Circ
  holds LE p1,p2,rectangle(-1,1,-1,1) & LE p2,p3,rectangle(-1,1,-1,1) &
  LE p3,p4,rectangle(-1,1,-1,1) implies
  f.p1,f.p2,f.p3,f.p4 are_in_this_order_on P
proof
  let p1,p2,p3,p4 be Point of TOP-REAL 2,
  P be non empty compact Subset of TOP-REAL 2,
  f be Function of TOP-REAL 2,TOP-REAL 2;
  set K = rectangle(-1,1,-1,1);
  assume that
A1: P= circle(0,0,1) and
A2: f=Sq_Circ;
A3: K is being_simple_closed_curve by Th50;
A4: K= {p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
  p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1} by Lm15;
A5: P={p: |.p.|=1} by A1,Th24;
  thus LE p1,p2,K & LE p2,p3,K & LE p3,p4,K implies
  f.p1,f.p2,f.p3,f.p4 are_in_this_order_on P
  proof
    assume that
A6: LE p1,p2,K and
A7: LE p2,p3,K and
A8: LE p3,p4,K;
A9: p1 in K by A3,A6,JORDAN7:5;
A10: p2 in K by A3,A6,JORDAN7:5;
A11: p3 in K by A3,A7,JORDAN7:5;
A12: p4 in K by A3,A8,JORDAN7:5;
    then
A13: ex q8 being Point of TOP-REAL 2 st ( q8=p4)&( q8`1=-1 & -1
<=q8`2 & q8`2<=1 or q8`2=1 & -1<=q8`1 & q8`1<=1 or q8 `1=1 & -1 <=q8`2 & q8`2<=
    1 or q8`2=-1 & -1<=q8`1 & q8`1<=1) by A4;
A14: LE p1,p3,K by A6,A7,Th50,JORDAN6:58;
A15: LE p2,p4,K by A7,A8,Th50,JORDAN6:58;
A16: W-min(K)=|[-1,-1]| by Th46;
A17: (|[-1,0]|)`2=0 by EUCLID:52;
A18: 1/2*(|[-1,-1]|+|[-1,1]|)=1/2*(|[-1,-1]|)+1/2*(|[-1,1]|) by RLVECT_1:def 5
      .= (|[1/2*(-1),1/2*(-1)]|)+1/2*(|[-1,1]|) by EUCLID:58
      .= (|[1/2*(-1),1/2*(-1)]|)+(|[1/2*(-1),1/2*1]|) by EUCLID:58
      .= (|[1/2*(-1)+1/2*(-1),1/2*(-1)+1/2*1]|) by EUCLID:56
      .= (|[(-1),0]|);
    then
A19: |[-1,0]| in LSeg(|[-1,-1]|,|[-1,1]|) by RLTOPSP1:69;
    now per cases by A9,A16,Th63,RLTOPSP1:68;
      case
A20:    p1 in LSeg(|[-1,-1]|,|[-1,1]|);
        then
A21:    p1`1=-1 by Th1;
        then
A22:    (f.p1)`1<0 by A2,Th68;
A23:    f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A24:    dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
        then
A25:    f.p1 in P by A9,A23,FUNCT_1:def 6;
        now per cases;
          case
A26:        p1`2>=0;
            then
A27:        LE f.p1,f.p2,P by A1,A2,A6,A20,Th65;
A28:        LE f.p2,f.p3,P by A1,A2,A6,A7,A20,A26,Th66;
            LE f.p3,f.p4,P by A1,A2,A8,A14,A20,A26,Th66;
            hence thesis by A27,A28,JORDAN17:def 1;
          end;
          case
A29:        p1`2<0;
            now per cases;
              case
A30:            p2`2<0 & p2 in LSeg(|[-1,-1]|,|[-1,1]|);
                then
A31:            p2`1=-1 by Th1;
A32:            f.p2 in P by A10,A23,A24,FUNCT_1:def 6;
A33:            p1`2<=p2`2 by A6,A20,A30,Th55;
                now per cases;
                  case
A34:                p3`2<0 & p3 in LSeg(|[-1,-1]|,|[-1,1]|);
                    then
A35:                p3`1=-1 by Th1;
A36:                f.p3 in P by A11,A23,A24,FUNCT_1:def 6;
A37:                p2`2<=p3`2 by A7,A30,A34,Th55;
                    now per cases;
                      case
A38:                    p4`2<0 & p4 in LSeg(|[-1,-1]|,|[-1,1]|);
                        then
A39:                    p4`1=-1 by Th1;
A40:                    (f.p2)`1<0 by A2,A30,A31,Th67;
A41:                    (f.p2)`2<0 by A2,A30,A31,Th67;
A42:                    (f.p3)`1<0 by A2,A34,A35,Th67;
A43:                    (f.p3)`2<0 by A2,A34,A35,Th67;
A44:                    (f.p4)`1<0 by A2,A38,A39,Th67;
A45:                    (f.p4)`2<0 by A2,A38,A39,Th67;
                        (f.p1)`2<=(f.p2)`2 by A2,A20,A30,A33,Th71;
                        then
A46:                    LE f.p1,f.p2,P by A5,A22,A25,A32,A40,A41,JGRAPH_5:51;
                        (f.p2)`2<=(f.p3)`2 by A2,A30,A34,A37,Th71;
                        then
A47:                    LE f.p2,f.p3,P by A5,A32,A36,A40,A42,A43,JGRAPH_5:51;
A48:                    f.p4 in P by A12,A23,A24,FUNCT_1:def 6;
                        p3`2<=p4`2 by A8,A34,A38,Th55;
                        then (f.p3)`2<=(f.p4)`2 by A2,A34,A38,Th71;
                        then LE f.p3,f.p4,P by A5,A36,A42,A44,A45,A48,
JGRAPH_5:51;
                        hence thesis by A46,A47,JORDAN17:def 1;
                      end;
                      case
A49:                    not(p4`2<0 & p4 in LSeg(|[-1,-1]|,|[-1,1]|));
A50:                    now per cases by A12,Th63;
                          case p4 in LSeg(|[-1,-1]|,|[-1,1]|);
hence p4 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p4`2
or p4 in LSeg(|[-1,1]|,|[1,1]|) or p4 in LSeg(|[1,1]|,|[1,-1]|)
or p4 in LSeg(|[1,-1]|,|[-1,-1]|) & p4<>W-min(K) by A49,EUCLID:52;
                          end;
                          case p4 in LSeg(|[-1,1]|,|[1,1]|);
hence p4 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p4`2
or p4 in LSeg(|[-1,1]|,|[1,1]|) or p4 in LSeg(|[1,1]|,|[1,-1]|)
                            or p4 in LSeg(|[1,-1]|,|[-1,-1]|) & p4<>W-min(K);
                          end;
                          case p4 in LSeg(|[1,1]|,|[1,-1]|);
hence p4 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p4`2
or p4 in LSeg(|[-1,1]|,|[1,1]|) or p4 in LSeg(|[1,1]|,|[1,-1]|)
                            or p4 in LSeg(|[1,-1]|,|[-1,-1]|) & p4<>W-min(K);
                          end;
                          case
A51:                        p4 in LSeg(|[1,-1]|,|[-1,-1]|);
A52:                        W-min(K)=|[-1,-1]| by Th46;
                            now
                              assume
A53:                          p4= W-min(K);
                              then p4`2=-1 by A52,EUCLID:52;
                              hence contradiction by A49,A52,A53,RLTOPSP1:68;
                            end;
hence p4 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p4`2
or p4 in LSeg(|[-1,1]|,|[1,1]|) or p4 in LSeg(|[1,1]|,|[1,-1]|)
or p4 in LSeg(|[1,-1]|,|[-1,-1]|) & p4<>W-min(K) by A51;
                          end;
                        end;
A54:                    (f.p2)`1<0 by A2,A30,A31,Th67;
A55:                    (f.p2)`2<0 by A2,A30,A31,Th67;
A56:                    (f.p3)`1<0 by A2,A34,A35,Th67;
A57:                    (f.p3)`2<0 by A2,A34,A35,Th67;
                        (f.p1)`2<=(f.p2)`2 by A2,A20,A30,A33,Th71;
                        then
A58:                    LE f.p1,f.p2,P by A5,A22,A25,A32,A54,A55,JGRAPH_5:51;
                        (f.p2)`2<=(f.p3)`2 by A2,A30,A34,A37,Th71;
                        then
A59:                    LE f.p2,f.p3,P by A5,A32,A36,A54,A56,A57,JGRAPH_5:51;
A60:                    now per cases;
                          case
A61:                        p4`1=-1 & p4`2<0 & p1`2<=p4`2;
                            now per cases by A50;
case p4 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p4`2;
                                hence contradiction by A61,EUCLID:52;
                              end;
                              case p4 in LSeg(|[-1,1]|,|[1,1]|);
                                hence contradiction by A61,Th3;
                              end;
                              case p4 in LSeg(|[1,1]|,|[1,-1]|);
                                hence contradiction by A61,Th1;
                              end;
                              case
                                A62:                            p4
 in LSeg(|[1,-1]|,|[-1,-1]|) & p4<>W-min(K);
                                then
A63:                            p4`2= -1 by Th3;
A64:                            W-min(K)= |[-1,-1]| by Th46;
                                then
A65:                            (W-min(K))`1=-1 by EUCLID:52;
                                (W-min(K))`2=-1 by A64,EUCLID:52;
                                hence contradiction by A61,A62,A63,A65,
TOPREAL3:6;
                              end;
                            end;
                            hence contradiction;
                          end;
                          case
A66:                        not (p4`1=-1 & p4`2<0 & p1`2<=p4`2);
A67:                        p4
in LSeg(|[-1,-1]|,|[-1,1]|) or p4 in LSeg(|[-1,1]|,|[1,1]|)
or p4 in LSeg(|[1,1]|,|[1,-1]|) or p4 in LSeg(|[1,-1]|,|[-1,-1]|) by A12,Th63;
                            now per cases by A66;
                              case
A68:                            p4`1<>-1;
A69:                            f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A70:                            dom
                                f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A71:                            (f.p1)`2<=0 by A2,A21,A29,Th67;
A72:                            Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A5,JGRAPH_5:34;
A73:                            f.p1 in P by A9,A69,A70,FUNCT_1:def 6;
                                Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A5,JGRAPH_5:35;
                                then
                                A74:                            f
.p1 in Lower_Arc(P) by A71,A73;
A75:                            f
                                .(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A76:                            now
                                  assume f.p1=W-min(P);
                                  then p1=|[-1,0]| by A2,A70,A75,FUNCT_1:def 4;
                                  hence contradiction by A29,EUCLID:52;
                                end;
                                now per cases by A67,A68,Th1;
                                  case p4 in LSeg(|[-1,1]|,|[1,1]|);
                                    then
                                    A77:                                p4
 `2=1 by Th3;
A78:                                f.p4 in P by A12,A69,A70,FUNCT_1:def 6;
                                    (f.p4)`2>=0 by A2,A77,Th69;
                                    then f.p4 in Upper_Arc(P) by A72,A78;
hence LE f.p4,f.p1,P by A74,A76,JORDAN6:def 10;
                                  end;
                                  case p4 in LSeg(|[1,1]|,|[1,-1]|);
                                    then
                                    A79:                                p4
 `1=1 by Th1;
A80:                                f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A81:                                dom
f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
                                    then
                                    A82:                                f
. p4 in P by A12,A80,FUNCT_1:def 6;
A83:                                f.p1 in P by A9,A80,A81,FUNCT_1:def 6;
A84:                                ( f.p1)`1<0 by A2,A21,A29,Th67;
A85:                                (f.p1)`2<=0 by A2,A21,A29,Th67;
A86:                                f
.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A87:                                now
                                      assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A81,A86,FUNCT_1:def 4;
                                      hence contradiction by A29,EUCLID:52;
                                    end;
A88:                                (f.p4)`1>=0 by A2,A79,Th68;
                                    now per cases;
                                      case
A89:                                    (f.p4)`2>=0;
A90:                                    (f.p1)`2<=0 by A2,A21,A29,Th67;
A91:                                    Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A5,JGRAPH_5:34;
A92:                                    f
.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A93:                                    now
                                          assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A81,A92,FUNCT_1:def 4;
                                          hence contradiction by A29,EUCLID:52;
                                        end;
A94:                                    f.p4 in Upper_Arc(P) by A82,A89,A91;
                                        Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A5,JGRAPH_5:35;
                                        then f.p1 in Lower_Arc(P) by A83,A90;
hence LE f.p4,f.p1,P by A93,A94,JORDAN6:def 10;
                                      end;
                                      case (f.p4)`2<0;
                                        hence LE f.p4,f.p1,P
                                        by A5,A82,A83,A84,A85,A87,A88,
JGRAPH_5:56;
                                      end;
                                    end;
                                    hence LE f.p4,f.p1,P;
                                  end;
                                  case
A95:                                p4 in LSeg(|[1,-1]|,|[-1,-1]|);
                                    then p4`2=-1 by Th3;
                                    then
                                    A96:                                (
f .p4)`2<0 by A2,Th69;
A97:                                f.p4 in P by A12,A69,A70,FUNCT_1:def 6;
A98:                                (f.p1)`2<=0 by A2,A21,A29,Th67;
                                    (f.p4)`1>=(f.p1)`1 by A2,A20,A95,Th70;
hence LE f.p4,f.p1,P by A5,A73,A76,A96,A97,A98,JGRAPH_5:56;
                                  end;
                                end;
                                hence LE f.p4,f.p1,P;
                              end;
                              case
A99:                            p4`1=-1 & p4`2>=0;
A100:                           f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A101:                           dom
                                f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
                                then
A102:                           f.p4 in P by A12,A100,FUNCT_1:def 6;
A103:                           f.p1 in P by A9,A100,A101,FUNCT_1:def 6;
A104:                           (f.p4)`2>=0 by A2,A99,Th69;
A105:                           (f.p1)`2<=0 by A2,A21,A29,Th67;
A106:                           Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A5,JGRAPH_5:34;
A107:                           f
                                .(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A108:                           now
                                  assume f.p1=W-min(P);
                                  then p1=|[-1,0]| by A2,A101,A107,
FUNCT_1:def 4;
                                  hence contradiction by A29,EUCLID:52;
                                end;
A109:                           f.p4 in Upper_Arc(P) by A102,A104,A106;
                                Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A5,JGRAPH_5:35;
                                then f.p1 in Lower_Arc(P) by A103,A105;
                                hence LE f.p4,f.p1,P by A108,A109,
JORDAN6:def 10;
                              end;
                              case
A110:                           p4`1=-1 & p4`2<0 & p1`2>p4`2;
                                then
                                A111:                           p4
 in LSeg (|[-1,-1]|,|[-1,1]|) by A13,Th2;
A112:                           f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A113:                           dom
                                f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
                                then
A114:                           f.p4 in P by A12,A112,FUNCT_1:def 6;
A115:                           f.p1 in P by A9,A112,A113,FUNCT_1:def 6;
A116:                           (f.p1)`1<0 by A2,A21,A29,Th67;
A117:                           (f.p1)`2<0 by A2,A21,A29,Th67;
A118:                           (f.p4)`2<=(f.p1)`2 by A2,A20,A29,A110,A111,Th71
                                ;
                                (f.p4)`1<0 by A2,A110,Th68;
hence LE f.p4,f.p1,P by A5,A114,A115,A116,A117,A118,JGRAPH_5:51;
                              end;
                            end;
                            hence LE f.p4,f.p1,P;
                          end;
                        end;
A119:                   K
={p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
                        p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1}
                        by Lm15;
                        thus K={p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                        1=p`2 & -1<=p`1 & p`1<=1}
                        proof
                          thus
                          K c= {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                          1=p`2 & -1<=p`1 & p`1<=1}
                          proof
                            let x be object;
                            assume x in K;
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1
<=p`1 & p`1<=1 or p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1) by
A119;
                            hence thesis;
                          end;
                          thus
                          {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                          1=p`2 & -1<=p`1 & p`1<=1} c= K
                          proof
                            let x be object;
                            assume x in {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                            1=p`2 & -1<=p`1 & p`1<=1};
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`1=1 & -1
<=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or 1=p`2 & -1<=p`1 & p`1<=1);
                            hence thesis by A119;
                          end;
                        end;
                        thus thesis by A58,A59,A60,JORDAN17:def 1;
                      end;
                    end;
                    hence thesis;
                  end;
                  case
A120:               not(p3`2<0 & p3 in LSeg(|[-1,-1]|,|[-1,1]|));
A121:               now per cases by A11,Th63;
                      case p3 in LSeg(|[-1,-1]|,|[-1,1]|);
hence p3 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p3`2
or p3 in LSeg(|[-1,1]|,|[1,1]|) or p3 in LSeg(|[1,1]|,|[1,-1]|)
or p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K) by A120,EUCLID:52;
                      end;
                      case p3 in LSeg(|[-1,1]|,|[1,1]|);
hence p3 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p3`2
or p3 in LSeg(|[-1,1]|,|[1,1]|) or p3 in LSeg(|[1,1]|,|[1,-1]|)
                        or p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K);
                      end;
                      case p3 in LSeg(|[1,1]|,|[1,-1]|);
hence p3 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p3`2
or p3 in LSeg(|[-1,1]|,|[1,1]|) or p3 in LSeg(|[1,1]|,|[1,-1]|)
                        or p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K);
                      end;
                      case
A122:                   p3 in LSeg(|[1,-1]|,|[-1,-1]|);
A123:                   W-min(K)=|[-1,-1]| by Th46;
                        now
                          assume
A124:                     p3= W-min(K);
                          then p3`2=-1 by A123,EUCLID:52;
                          hence contradiction by A120,A123,A124,RLTOPSP1:68;
                        end;
hence p3 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p3`2
or p3 in LSeg(|[-1,1]|,|[1,1]|) or p3 in LSeg(|[1,1]|,|[1,-1]|)
or p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K) by A122;
                      end;
                    end;
                    then
A125:               LE |[-1,0]|,p3,K by A19,Th59;
A126:               (f.p2)`1<0 by A2,A30,A31,Th67;
A127:               (f.p2)`2<0 by A2,A30,A31,Th67;
                    (f.p1)`2<=(f.p2)`2 by A2,A20,A30,A33,Th71;
                    then
A128:               LE f.p1,f.p2,P by A5,A22,A25,A32,A126,A127,JGRAPH_5:51;
A129:               LE f.p3,f.p4,P by A1,A2,A8,A17,A18,A125,Th66,RLTOPSP1:69;
A130:               now per cases;
                      case
A131:                   p4`1=-1 & p4`2<0 & p1`2<=p4`2;
A132:                   (|[-1,-1]|)`1=-1 by EUCLID:52;
A133:                   (|[-1,-1]|)`2=-1 by EUCLID:52;
A134:                   (|[-1,1]|)`1=-1 by EUCLID:52;
A135:                   (|[-1,1]|)`2=1 by EUCLID:52;
                        -1<=p4`2 by A12,Th19;
                        then
A136:                   p4
in LSeg(|[-1,-1]|,|[-1,1]|) by A131,A132,A133,A134,A135,GOBOARD7:7;
                        now per cases by A121;
                          case
A137:                       p3
                            in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p3`2;
                            then 0<=p3`2 by EUCLID:52;
                            hence contradiction by A8,A131,A136,A137,Th55;
                          end;
                          case
A138:                       p3 in LSeg(|[-1,1]|,|[1,1]|);
                            then LE p4,p3,K by A136,Th59;
                            then p3=p4 by A8,Th50,JORDAN6:57;
                            hence contradiction by A131,A138,Th3;
                          end;
                          case
A139:                       p3 in LSeg(|[1,1]|,|[1,-1]|);
                            then LE p4,p3,K by A136,Th59;
                            then p3=p4 by A8,Th50,JORDAN6:57;
                            hence contradiction by A131,A139,Th1;
                          end;
                          case
A140:                       p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K);
                            then LE p4,p3,K by A136,Th59;
                            then
A141:                       p3=p4 by A8,Th50,JORDAN6:57;
A142:                       p3`2= -1 by A140,Th3;
A143:                       W-min(K)= |[-1,-1]| by Th46;
                            then
A144:                       (W-min(K))`1=-1 by EUCLID:52;
                            (W-min(K))`2=-1 by A143,EUCLID:52;
hence contradiction by A131,A140,A141,A142,A144,TOPREAL3:6;
                          end;
                        end;
                        hence contradiction;
                      end;
                      case
A145:                   not (p4`1=-1 & p4`2<0 & p1`2<=p4`2);
A146:                   p4
in LSeg(|[-1,-1]|,|[-1,1]|) or p4 in LSeg(|[-1,1]|,|[1,1]|)
or p4 in LSeg(|[1,1]|,|[1,-1]|) or p4 in LSeg(|[1,-1]|,|[-1,-1]|) by A12,Th63;
                        now per cases by A145;
                          case
A147:                       p4`1<>-1;
A148:                       f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
                            A149:                       dom
 f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A150:                       (f.p1)`2<=0 by A2,A21,A29,Th67;
A151:                       Upper_Arc(P)
                            ={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
                            by A5,JGRAPH_5:34;
A152:                       f.p1 in P by A9,A148,A149,FUNCT_1:def 6;
                            Lower_Arc(P)
                            ={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
                            by A5,JGRAPH_5:35;
                            then
                            A153:                       f
.p1 in Lower_Arc( P) by A150,A152;
A154:                       f.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A155:                       now
                              assume f.p1=W-min(P);
                              then p1=|[-1,0]| by A2,A149,A154,FUNCT_1:def 4;
                              hence contradiction by A29,EUCLID:52;
                            end;
                            now per cases by A146,A147,Th1;
                              case p4 in LSeg(|[-1,1]|,|[1,1]|);
                                then
A156:                           p4`2=1 by Th3;
A157:                           f.p4 in P by A12,A148,A149,FUNCT_1:def 6;
                                (f.p4)`2>=0 by A2,A156,Th69;
                                then f.p4 in Upper_Arc(P) by A151,A157;
hence LE f.p4,f.p1,P by A153,A155,JORDAN6:def 10;
                              end;
                              case p4 in LSeg(|[1,1]|,|[1,-1]|);
                                then
A158:                           p4`1=1 by Th1;
A159:                           f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A160:                           dom
                                f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
                                then
A161:                           f.p4 in P by A12,A159,FUNCT_1:def 6;
A162:                           f.p1 in P by A9,A159,A160,FUNCT_1:def 6;
A163:                           (f.p1)`1<0 by A2,A21,A29,Th67;
A164:                           (f.p1)`2<=0 by A2,A21,A29,Th67;
A165:                           f
                                .(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A166:                           now
                                  assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A160,A165,FUNCT_1:def 4;
                                  hence contradiction by A29,EUCLID:52;
                                end;
A167:                           (f.p4)`1>=0 by A2,A158,Th68;
                                now per cases;
                                  case
A168:                               (f.p4)`2>=0;
A169:                               (f.p1)`2<=0 by A2,A21,A29,Th67;
A170:                               Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A5,JGRAPH_5:34;
A171:                               f
.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A172:                               now
                                      assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A160,A171,FUNCT_1:def 4;
                                      hence contradiction by A29,EUCLID:52;
                                    end;
A173:                               f.p4 in Upper_Arc(P) by A161,A168,A170;
                                    Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A5,JGRAPH_5:35;
                                    then f.p1 in Lower_Arc(P) by A162,A169;
hence LE f.p4,f.p1,P by A172,A173,JORDAN6:def 10;
                                  end;
                                  case (f.p4)`2<0;
                                    hence LE f.p4,f.p1,P
                                    by A5,A161,A162,A163,A164,A166,A167,
JGRAPH_5:56;
                                  end;
                                end;
                                hence LE f.p4,f.p1,P;
                              end;
                              case
A174:                           p4 in LSeg(|[1,-1]|,|[-1,-1]|);
                                then p4`2=-1 by Th3;
                                then
A175:                           (f.p4)`2<0 by A2,Th69;
A176:                           f.p4 in P by A12,A148,A149,FUNCT_1:def 6;
A177:                           (f.p1)`2<=0 by A2,A21,A29,Th67;
                                (f.p4)`1>=(f.p1)`1 by A2,A20,A174,Th70;
hence LE f.p4,f.p1,P by A5,A152,A155,A175,A176,A177,JGRAPH_5:56;
                              end;
                            end;
                            hence LE f.p4,f.p1,P;
                          end;
                          case
A178:                       p4`1=-1 & p4`2>=0;
A179:                       f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
                            A180:                       dom
 f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
                            then
A181:                       f.p4 in P by A12,A179,FUNCT_1:def 6;
A182:                       f.p1 in P by A9,A179,A180,FUNCT_1:def 6;
A183:                       (f.p4)`2>=0 by A2,A178,Th69;
A184:                       (f.p1)`2<=0 by A2,A21,A29,Th67;
A185:                       Upper_Arc(P)
                            ={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
                            by A5,JGRAPH_5:34;
A186:                       f.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A187:                       now
                              assume f.p1=W-min(P);
                              then p1=|[-1,0]| by A2,A180,A186,FUNCT_1:def 4;
                              hence contradiction by A29,EUCLID:52;
                            end;
A188:                       f.p4 in Upper_Arc(P) by A181,A183,A185;
                            Lower_Arc(P)
                            ={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
                            by A5,JGRAPH_5:35;
                            then f.p1 in Lower_Arc(P) by A182,A184;
                            hence LE f.p4,f.p1,P by A187,A188,JORDAN6:def 10;
                          end;
                          case
A189:                       p4`1=-1 & p4`2<0 & p1`2>p4`2;
                            then
                            A190:                       p4
 in LSeg(|[-1,-1 ]|,|[-1,1]|) by A13,Th2;
A191:                       f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
                            A192:                       dom
 f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
                            then
A193:                       f.p4 in P by A12,A191,FUNCT_1:def 6;
A194:                       f.p1 in P by A9,A191,A192,FUNCT_1:def 6;
A195:                       (f.p1)`1<0 by A2,A21,A29,Th67;
A196:                       (f.p1)`2<0 by A2,A21,A29,Th67;
A197:                       (f.p4)`2<=(f.p1)`2 by A2,A20,A29,A189,A190,Th71;
                            (f.p4)`1<0 by A2,A189,Th68;
hence LE f.p4,f.p1,P by A5,A193,A194,A195,A196,A197,JGRAPH_5:51;
                          end;
                        end;
                        hence LE f.p4,f.p1,P;
                      end;
                    end;
A198:               K
={p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
                    p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1}
                    by Lm15;
                    thus K={p: p`1=-1 & -1 <=p`2 & p`2<=1 or
                    p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                    1=p`2 & -1<=p`1 & p`1<=1}
                    proof
                      thus K c= {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
                      p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                      1=p`2 & -1<=p`1 & p`1<=1}
                      proof
                        let x be object;
                        assume x in K;
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1
<=p`1 & p`1<=1 or p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1) by
A198;
                        hence thesis;
                      end;
                      thus
                      {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
                      p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                      1=p`2 & -1<=p`1 & p`1<=1} c= K
                      proof
                        let x be object;
                        assume x in {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                        1=p`2 & -1<=p`1 & p`1<=1};
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`1=1 & -1
<=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or 1=p`2 & -1<=p`1 & p`1<=1);
                        hence thesis by A198;
                      end;
                    end;
                    thus thesis by A128,A129,A130,JORDAN17:def 1;
                  end;
                end;
                hence thesis;
              end;
              case
A199:           not(p2`2<0 & p2 in LSeg(|[-1,-1]|,|[-1,1]|));
A200:           now per cases by A10,Th63;
                  case p2 in LSeg(|[-1,-1]|,|[-1,1]|);
                    hence
                    p2 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p2`2
or p2 in LSeg(|[-1,1]|,|[1,1]|) or p2 in LSeg(|[1,1]|,|[1,-1]|)
or p2 in LSeg(|[1,-1]|,|[-1,-1]|) & p2<>W-min(K) by A199,EUCLID:52;
                  end;
                  case p2 in LSeg(|[-1,1]|,|[1,1]|);
                    hence
                    p2 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p2`2
or p2 in LSeg(|[-1,1]|,|[1,1]|) or p2 in LSeg(|[1,1]|,|[1,-1]|)
                    or p2 in LSeg(|[1,-1]|,|[-1,-1]|) & p2<>W-min(K);
                  end;
                  case p2 in LSeg(|[1,1]|,|[1,-1]|);
                    hence
                    p2 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p2`2
or p2 in LSeg(|[-1,1]|,|[1,1]|) or p2 in LSeg(|[1,1]|,|[1,-1]|)
                    or p2 in LSeg(|[1,-1]|,|[-1,-1]|) & p2<>W-min(K);
                  end;
                  case
A201:               p2 in LSeg(|[1,-1]|,|[-1,-1]|);
A202:               W-min(K)=|[-1,-1]| by Th46;
                    now
                      assume
A203:                 p2= W-min(K);
                      then p2`2=-1 by A202,EUCLID:52;
                      hence contradiction by A199,A202,A203,RLTOPSP1:68;
                    end;
                    hence
                    p2 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p2`2
or p2 in LSeg(|[-1,1]|,|[1,1]|) or p2 in LSeg(|[1,1]|,|[1,-1]|)
                    or p2 in LSeg(|[1,-1]|,|[-1,-1]|) & p2<>W-min(K) by A201;
                  end;
                end;
                then
A204:           LE |[-1,0]|,p2,K by A19,Th59;
                then
A205:           LE f.p2,f.p3,P by A1,A2,A7,A17,A18,Th66,RLTOPSP1:69;
                LE |[-1,0]|,p3,K by A7,A204,Th50,JORDAN6:58;
                then
A206:           LE f.p3,f.p4,P by A1,A2,A8,A17,A18,Th66,RLTOPSP1:69;
A207:           now per cases;
                  case
A208:               p4`1=-1 & p4`2<0 & p1`2<=p4`2;
A209:               (|[-1,-1]|)`1=-1 by EUCLID:52;
A210:               (|[-1,-1]|)`2=-1 by EUCLID:52;
A211:               (|[-1,1]|)`1=-1 by EUCLID:52;
A212:               (|[-1,1]|)`2=1 by EUCLID:52;
                    -1<=p4`2 by A12,Th19;
                    then
A213:               p4 in LSeg(|[-1,-1]|,|[-1,1]|) by A208,A209,A210,A211,A212,
GOBOARD7:7;
                    now per cases by A200;
                      case
A214:                   p2 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p2`2;
                        then 0<=p2`2 by EUCLID:52;
                        hence contradiction by A15,A208,A213,A214,Th55;
                      end;
                      case
A215:                   p2 in LSeg(|[-1,1]|,|[1,1]|);
                        then LE p4,p2,K by A213,Th59;
                        then p2=p4 by A15,Th50,JORDAN6:57;
                        hence contradiction by A208,A215,Th3;
                      end;
                      case
A216:                   p2 in LSeg(|[1,1]|,|[1,-1]|);
                        then LE p4,p2,K by A213,Th59;
                        then p2=p4 by A15,Th50,JORDAN6:57;
                        hence contradiction by A208,A216,Th1;
                      end;
                      case
A217:                   p2 in LSeg(|[1,-1]|,|[-1,-1]|) & p2<>W-min(K);
                        then LE p4,p2,K by A213,Th59;
                        then
A218:                   p2=p4 by A15,Th50,JORDAN6:57;
A219:                   p2`2= -1 by A217,Th3;
A220:                   W-min(K)= |[-1,-1]| by Th46;
                        then
A221:                   (W-min(K))`1=-1 by EUCLID:52;
                        (W-min(K))`2=-1 by A220,EUCLID:52;
                        hence contradiction by A208,A217,A218,A219,A221,
TOPREAL3:6;
                      end;
                    end;
                    hence contradiction;
                  end;
                  case
A222:               not (p4`1=-1 & p4`2<0 & p1`2<=p4`2);
A223:               p4
                    in LSeg(|[-1,-1]|,|[-1,1]|) or p4 in LSeg(|[-1,1]|,|[1,1]|)
or p4 in LSeg(|[1,1]|,|[1,-1]|) or p4 in LSeg(|[1,-1]|,|[-1,-1]|) by A12,Th63;
                    now per cases by A222;
                      case
A224:                   p4`1<>-1;
A225:                   f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A226:                   dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A227:                   (f.p1)`2<=0 by A2,A21,A29,Th67;
A228:                   Upper_Arc(P)
                        ={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
                        by A5,JGRAPH_5:34;
A229:                   f.p1 in P by A9,A225,A226,FUNCT_1:def 6;
                        Lower_Arc(P)
                        ={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
                        by A5,JGRAPH_5:35;
                        then
A230:                   f.p1 in Lower_Arc(P) by A227,A229;
A231:                   f.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A232:                   now
                          assume f.p1=W-min(P);
                          then p1=|[-1,0]| by A2,A226,A231,FUNCT_1:def 4;
                          hence contradiction by A29,EUCLID:52;
                        end;
                        now per cases by A223,A224,Th1;
                          case p4 in LSeg(|[-1,1]|,|[1,1]|);
                            then
A233:                       p4`2=1 by Th3;
A234:                       f.p4 in P by A12,A225,A226,FUNCT_1:def 6;
                            (f.p4)`2>=0 by A2,A233,Th69;
                            then f.p4 in Upper_Arc(P) by A228,A234;
                            hence LE f.p4,f.p1,P by A230,A232,JORDAN6:def 10;
                          end;
                          case p4 in LSeg(|[1,1]|,|[1,-1]|);
                            then
A235:                       p4`1=1 by Th1;
A236:                       f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
                            A237:                       dom
 f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
                            then
A238:                       f.p4 in P by A12,A236,FUNCT_1:def 6;
A239:                       f.p1 in P by A9,A236,A237,FUNCT_1:def 6;
A240:                       (f.p1)`1<0 by A2,A21,A29,Th67;
A241:                       (f.p1)`2<=0 by A2,A21,A29,Th67;
A242:                       f.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A243:                       now
                              assume f.p1=W-min(P);
                              then p1=|[-1,0]| by A2,A237,A242,FUNCT_1:def 4;
                              hence contradiction by A29,EUCLID:52;
                            end;
A244:                       (f.p4)`1>=0 by A2,A235,Th68;
                            now per cases;
                              case
A245:                           (f.p4)`2>=0;
A246:                           (f.p1)`2<=0 by A2,A21,A29,Th67;
A247:                           Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A5,JGRAPH_5:34;
A248:                           f
                                .(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A249:                           now
                                  assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A237,A248,FUNCT_1:def 4;
                                  hence contradiction by A29,EUCLID:52;
                                end;
A250:                           f.p4 in Upper_Arc(P) by A238,A245,A247;
                                Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A5,JGRAPH_5:35;
                                then f.p1 in Lower_Arc(P) by A239,A246;
hence LE f.p4,f.p1,P by A249,A250,JORDAN6:def 10;
                              end;
                              case (f.p4)`2<0;
                                hence LE f.p4,f.p1,P
                                by A5,A238,A239,A240,A241,A243,A244,JGRAPH_5:56
;
                              end;
                            end;
                            hence LE f.p4,f.p1,P;
                          end;
                          case
A251:                       p4 in LSeg(|[1,-1]|,|[-1,-1]|);
                            then p4`2=-1 by Th3;
                            then
A252:                       (f.p4)`2<0 by A2,Th69;
A253:                       f.p4 in P by A12,A225,A226,FUNCT_1:def 6;
A254:                       (f.p1)`2<=0 by A2,A21,A29,Th67;
                            (f.p4)`1>=(f.p1)`1 by A2,A20,A251,Th70;
hence LE f.p4,f.p1,P by A5,A229,A232,A252,A253,A254,JGRAPH_5:56;
                          end;
                        end;
                        hence LE f.p4,f.p1,P;
                      end;
                      case
A255:                   p4`1=-1 & p4`2>=0;
A256:                   f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A257:                   dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
                        then
A258:                   f.p4 in P by A12,A256,FUNCT_1:def 6;
A259:                   f.p1 in P by A9,A256,A257,FUNCT_1:def 6;
A260:                   (f.p4)`2>=0 by A2,A255,Th69;
A261:                   (f.p1)`2<=0 by A2,A21,A29,Th67;
A262:                   Upper_Arc(P)
                        ={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
                        by A5,JGRAPH_5:34;
A263:                   f.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A264:                   now
                          assume f.p1=W-min(P);
                          then p1=|[-1,0]| by A2,A257,A263,FUNCT_1:def 4;
                          hence contradiction by A29,EUCLID:52;
                        end;
A265:                   f.p4 in Upper_Arc(P) by A258,A260,A262;
                        Lower_Arc(P)
                        ={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
                        by A5,JGRAPH_5:35;
                        then f.p1 in Lower_Arc(P) by A259,A261;
                        hence LE f.p4,f.p1,P by A264,A265,JORDAN6:def 10;
                      end;
                      case
A266:                   p4`1=-1 & p4`2<0 & p1`2>p4`2;
                        then
                        A267:                   p4
 in LSeg(|[-1,-1]|,|[-1, 1]|) by A13,Th2;
A268:                   f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A269:                   dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
                        then
A270:                   f.p4 in P by A12,A268,FUNCT_1:def 6;
A271:                   f.p1 in P by A9,A268,A269,FUNCT_1:def 6;
A272:                   (f.p1)`1<0 by A2,A21,A29,Th67;
A273:                   (f.p1)`2<0 by A2,A21,A29,Th67;
A274:                   (f.p4)`2<=(f.p1)`2 by A2,A20,A29,A266,A267,Th71;
                        (f.p4)`1<0 by A2,A266,Th68;
hence LE f.p4,f.p1,P by A5,A270,A271,A272,A273,A274,JGRAPH_5:51;
                      end;
                    end;
                    hence LE f.p4,f.p1,P;
                  end;
                end;
                A275:           K
={p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
                p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1}
                by Lm15;
                thus K={p: p`1=-1 & -1 <=p`2 & p`2<=1 or
                p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                1=p`2 & -1<=p`1 & p`1<=1}
                proof
                  thus K c= {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
                  p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                  1=p`2 & -1<=p`1 & p`1<=1}
                  proof
                    let x be object;
                    assume x in K;
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1
<=p`1 & p`1<=1 or p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1) by
A275;
                    hence thesis;
                  end;
                  thus
                  {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
                  p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                  1=p`2 & -1<=p`1 & p`1<=1} c= K
                  proof
                    let x be object;
                    assume x in {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
                    p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
                    1=p`2 & -1<=p`1 & p`1<=1};
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`1=1 & -1
<=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or 1=p`2 & -1<=p`1 & p`1<=1);
                    hence thesis by A275;
                  end;
                end;
                thus thesis by A205,A206,A207,JORDAN17:def 1;
              end;
            end;
            hence thesis;
          end;
        end;
        hence thesis;
      end;
      case
A276:   p1 in LSeg(|[-1,1]|,|[1,1]|);
A277:   |[-1,1]| in LSeg(|[-1,1]|,|[1,1]|) by RLTOPSP1:68;
A278:   (|[-1,1]|)`1=-1 by EUCLID:52;
A279:   (|[-1,1]|)`2=1 by EUCLID:52;
        -1 <=p1`1 by A276,Th3;
        then
A280:   LE |[-1,1]|,p1,K by A276,A277,A278,Th60;
        then
A281:   LE f.p1,f.p2,P by A1,A2,A6,A279,Th66,RLTOPSP1:68;
A282:   LE |[-1,1]|,p2,K by A6,A280,Th50,JORDAN6:58;
        then
A283:   LE f.p2,f.p3,P by A1,A2,A7,A279,Th66,RLTOPSP1:68;
        LE |[-1,1]|,p3,K by A7,A282,Th50,JORDAN6:58;
        then LE f.p3,f.p4,P by A1,A2,A8,A279,Th66,RLTOPSP1:68;
        hence thesis by A281,A283,JORDAN17:def 1;
      end;
      case
A284:   p1 in LSeg(|[1,1]|,|[1,-1]|);
A285:   |[-1,1]| in LSeg(|[-1,1]|,|[1,1]|) by RLTOPSP1:68;
A286:   (|[-1,1]|)`1=-1 by EUCLID:52;
A287:   (|[-1,1]|)`2=1 by EUCLID:52;
A288:   |[1,1]| in LSeg(|[1,1]|,|[1,-1]|) by RLTOPSP1:68;
A289:   |[1,1]| in LSeg(|[-1,1]|,|[1,1]|) by RLTOPSP1:68;
A290:   (|[1,1]|)`1=1 by EUCLID:52;
A291:   (|[1,1]|)`2=1 by EUCLID:52;
A292:   LE |[-1,1]|,|[1,1]|,K by A285,A286,A289,A290,Th60;
        p1`2<=1 by A284,Th1;
        then LE |[1,1]|,p1,K by A284,A288,A291,Th61;
        then
A293:   LE |[-1,1]|,p1,K by A292,Th50,JORDAN6:58;
        then
A294:   LE f.p1,f.p2,P by A1,A2,A6,A287,Th66,RLTOPSP1:68;
A295:   LE |[-1,1]|,p2,K by A6,A293,Th50,JORDAN6:58;
        then
A296:   LE f.p2,f.p3,P by A1,A2,A7,A287,Th66,RLTOPSP1:68;
        LE |[-1,1]|,p3,K by A7,A295,Th50,JORDAN6:58;
        then LE f.p3,f.p4,P by A1,A2,A8,A287,Th66,RLTOPSP1:68;
        hence thesis by A294,A296,JORDAN17:def 1;
      end;
      case
A297:   p1 in LSeg(|[1,-1]|,|[-1,-1]|) & p1 <> W-min(K);
A298:   |[-1,1]| in LSeg(|[-1,1]|,|[1,1]|) by RLTOPSP1:68;
A299:   (|[-1,1]|)`1=-1 by EUCLID:52;
A300:   (|[-1,1]|)`2=1 by EUCLID:52;
A301:   |[1,1]| in LSeg(|[-1,1]|,|[1,1]|) by RLTOPSP1:68;
        (|[1,1]|)`1=1 by EUCLID:52;
        then
A302:   LE |[-1,1]|,|[1,1]|,K by A298,A299,A301,Th60;
A303:   |[1,-1]| in LSeg(|[1,1]|,|[1,-1]|) by RLTOPSP1:68;
A304:   |[1,-1]| in LSeg(|[1,-1]|,|[-1,-1]|) by RLTOPSP1:68;
A305:   (|[1,-1]|)`1=1 by EUCLID:52;
        LE |[1,1]|,|[1,-1]|,K by A301,A303,Th60;
        then
A306:   LE |[-1,1]|,|[1,-1]|,K by A302,Th50,JORDAN6:58;
        W-min(K)=|[-1,-1]| by Th46;
        then
A307:   (W-min(K))`1=-1 by EUCLID:52;
        p1`1<=1 by A297,Th3;
        then LE |[1,-1]|,p1,K by A297,A304,A305,A307,Th62;
        then
A308:   LE |[-1,1]|,p1,K by A306,Th50,JORDAN6:58;
        then
A309:   LE f.p1,f.p2,P by A1,A2,A6,A300,Th66,RLTOPSP1:68;
A310:   LE |[-1,1]|,p2,K by A6,A308,Th50,JORDAN6:58;
        then
A311:   LE f.p2,f.p3,P by A1,A2,A7,A300,Th66,RLTOPSP1:68;
        LE |[-1,1]|,p3,K by A7,A310,Th50,JORDAN6:58;
        then LE f.p3,f.p4,P by A1,A2,A8,A300,Th66,RLTOPSP1:68;
        hence thesis by A309,A311,JORDAN17:def 1;
      end;
    end;
    hence thesis;
  end;
end;
