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

theorem Th77:
  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 &
  LE f.p1,f.p2,P & LE f.p2,f.p3,P & LE f.p3,f.p4,P
  holds p1,p2,p3,p4 are_in_this_order_on rectangle(-1,1,-1,1)
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 and
A3: LE f.p1,f.p2,P and
A4: LE f.p2,f.p3,P and
A5: LE f.p3,f.p4,P;
A6: K is being_simple_closed_curve by Th50;
A7: P={p: |.p.|=1} by A1,Th24;
  then
A8: LE f.p1,f.p3,P by A3,A4,JGRAPH_3:26,JORDAN6:58;
A9: LE f.p2,f.p4,P by A4,A5,A7,JGRAPH_3:26,JORDAN6:58;
A10: f.:K=P by A2,A7,Lm15,Th35,JGRAPH_3:23;
A11: dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A12: P={p: |.p.|=1} by A1,Th24;
  then
A13: P is being_simple_closed_curve by JGRAPH_3:26;
  then f.p1 in P by A3,JORDAN7:5;
  then ex x1 being object st ( x1 in dom f)&( x1 in K)&( f.p1=f.x1)
  by A10,FUNCT_1:def 6;
  then
A14: p1 in K by A2,A11,FUNCT_1:def 4;
  f.p2 in P by A3,A13,JORDAN7:5;
  then ex x2 being object st ( x2 in dom f)&( x2 in K)&( f.p2=f.x2)
  by A10,FUNCT_1:def 6;
  then
A15: p2 in K by A2,A11,FUNCT_1:def 4;
  f.p3 in P by A4,A13,JORDAN7:5;
  then ex x3 being object st ( x3 in dom f)&( x3 in K)&( f.p3=f.x3)
  by A10,FUNCT_1:def 6;
  then
A16: p3 in K by A2,A11,FUNCT_1:def 4;
  f.p4 in P by A5,A13,JORDAN7:5;
  then ex x4 being object st ( x4 in dom f)&( x4 in K)&( f.p4=f.x4)
  by A10,FUNCT_1:def 6;
  then
A17: p4 in K by A2,A11,FUNCT_1:def 4;
  now
    assume
A18: not p1,p2,p3,p4 are_in_this_order_on K;
A19: now
      assume
A20:  p1,p2,p4,p3 are_in_this_order_on K;
      now per cases by A20,JORDAN17:def 1;
        case
A21:      LE p1,p2,K & LE p2,p4,K & LE p4,p3,K;
          then f.p1,f.p2,f.p4,f.p3 are_in_this_order_on P by A1,A2,Th72;
          then LE f.p1,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p3,P or
          LE f.p2,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p1,P or
          LE f.p4,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p2,P or
          LE f.p3,f.p1,P & LE f.p1,f.p2,P & LE f.p2,f.p4,P by JORDAN17:def 1;
          then f.p3=f.p4 or f.p3=f.p1 by A5,A8,A12,JGRAPH_3:26,JORDAN6:57;
          then
A22:      p3=p4 or p3=p1 by A2,A11,FUNCT_1:def 4;
          LE p1,p4,K by A21,Th50,JORDAN6:58;
          then p1=p4 by A18,A20,A21,A22,Th50,JORDAN6:57;
          hence contradiction by A18,A20,A22;
        end;
        case
A23:      LE p2,p4,K & LE p4,p3,K & LE p3,p1,K;
          then f.p2,f.p4,f.p3,f.p1 are_in_this_order_on P by A1,A2,Th72;
          then LE f.p2,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p1,P or
          LE f.p4,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p2,P or
          LE f.p3,f.p1,P & LE f.p1,f.p2,P & LE f.p2,f.p4,P or
          LE f.p1,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p3,P by JORDAN17:def 1;
          then f.p3=f.p4 or LE f.p3,f.p2,P by A5,A13,JORDAN6:57,58;
          then f.p3=f.p4 or f.p3=f.p2 by A4,A12,JGRAPH_3:26,JORDAN6:57;
          then
A24:      p3=p4 or p3=p2 by A2,A11,FUNCT_1:def 4;
          then p4=p2 by A18,A20,A23,Th50,JORDAN6:57;
          hence contradiction by A18,A20,A24;
        end;
        case
A25:      LE p4,p3,K & LE p3,p1,K & LE p1,p2,K;
          then f.p4,f.p3,f.p1,f.p2 are_in_this_order_on P by A1,A2,Th72;
          then LE f.p4,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p2,P or
          LE f.p3,f.p1,P & LE f.p1,f.p2,P & LE f.p2,f.p4,P or
          LE f.p1,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p3,P or
          LE f.p2,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p1,P by JORDAN17:def 1;
          then f.p3=f.p4 or LE f.p3,f.p2,P by A5,A13,JORDAN6:57,58;
          then f.p3=f.p4 or f.p3=f.p2 by A4,A12,JGRAPH_3:26,JORDAN6:57;
          then
A26:      p3=p4 or p3=p2 by A2,A11,FUNCT_1:def 4;
          then p3=p1 by A18,A20,A25,Th50,JORDAN6:57;
          hence contradiction by A6,A18,A20,A26,JORDAN17:12;
        end;
        case
A27:      LE p3,p1,K & LE p1,p2,K & LE p2,p4,K;
          then f.p3,f.p1,f.p2,f.p4 are_in_this_order_on P by A1,A2,Th72;
          then LE f.p4,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p2,P or
          LE f.p3,f.p1,P & LE f.p1,f.p2,P & LE f.p2,f.p4,P or
          LE f.p1,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p3,P or
          LE f.p2,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p1,P by JORDAN17:def 1;
          then f.p3=f.p4 or LE f.p3,f.p2,P by A5,A13,JORDAN6:57,58;
          then f.p3=f.p4 or f.p3=f.p2 by A4,A12,JGRAPH_3:26,JORDAN6:57;
          then
A28:      p3=p4 or p3=p2 by A2,A11,FUNCT_1:def 4;
          then p3=p1 by A18,A20,A27,Th50,JORDAN6:57;
          hence contradiction by A6,A18,A20,A28,JORDAN17:12;
        end;
      end;
      hence contradiction;
    end;
A29: now
      assume
A30:  p1,p3,p4,p2 are_in_this_order_on K;
      now per cases by A30,JORDAN17:def 1;
        case LE p1,p3,K & LE p3,p4,K & LE p4,p2,K;
          then f.p1,f.p3,f.p4,f.p2 are_in_this_order_on P by A1,A2,Th72;
          then LE f.p1,f.p3,P & LE f.p3,f.p4,P & LE f.p4,f.p2,P or
          LE f.p3,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p1,P or
          LE f.p4,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p3,P or
          LE f.p2,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p4,P by JORDAN17:def 1;
          then f.p4=f.p2 or f.p2=f.p1 by A3,A9,A12,JGRAPH_3:26,JORDAN6:57;
          then
A31:      p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
          then f.p3=f.p2 or f.p4=f.p1 by A4,A5,A6,A12,A18,A30,JGRAPH_3:26
,JORDAN17:12,JORDAN6:57;
          then p3=p2 or p4=p1 by A2,A11,FUNCT_1:def 4;
          hence contradiction by A6,A18,A30,A31,JORDAN17:12;
        end;
        case LE p3,p4,K & LE p4,p2,K & LE p2,p1,K;
          then f.p3,f.p4,f.p2,f.p1 are_in_this_order_on P by A1,A2,Th72;
          then
A32:      LE f.p1,f.p3,P & LE f.p3,f.p4,P & LE f.p4,f.p2,P or
          LE f.p3,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p1,P or
          LE f.p4,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p3,P or
          LE f.p2,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p4,P by JORDAN17:def 1;
          then f.p4=f.p2 or f.p2=f.p1 by A3,A9,A12,JGRAPH_3:26,JORDAN6:57;
          then
A33:      p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
          f.p2=f.p1 or LE f.p3,f.p2,P by A3,A13,A32,JORDAN6:57,58;
          then f.p2=f.p1 or f.p3=f.p2 by A4,A12,JGRAPH_3:26,JORDAN6:57;
          then p2=p1 or p3=p2 by A2,A11,FUNCT_1:def 4;
          hence contradiction by A6,A18,A30,A33,JORDAN17:12;
        end;
        case LE p4,p2,K & LE p2,p1,K & LE p1,p3,K;
          then f.p4,f.p2,f.p1,f.p3 are_in_this_order_on P by A1,A2,Th72;
          then
A34:      LE f.p1,f.p3,P & LE f.p3,f.p4,P & LE f.p4,f.p2,P or
          LE f.p3,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p1,P or
          LE f.p4,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p3,P or
          LE f.p2,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p4,P by JORDAN17:def 1;
          then f.p4=f.p2 or f.p2=f.p1 by A3,A9,A12,JGRAPH_3:26,JORDAN6:57;
          then
A35:      p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
          f.p2=f.p1 or LE f.p3,f.p2,P by A3,A13,A34,JORDAN6:57,58;
          then f.p2=f.p1 or f.p3=f.p2 by A4,A12,JGRAPH_3:26,JORDAN6:57;
          then p2=p1 or p3=p2 by A2,A11,FUNCT_1:def 4;
          hence contradiction by A6,A18,A30,A35,JORDAN17:12;
        end;
        case
A36:      LE p2,p1,K & LE p1,p3,K & LE p3,p4,K;
          then f.p2,f.p1,f.p3,f.p4 are_in_this_order_on P by A1,A2,Th72;
          then LE f.p1,f.p3,P & LE f.p3,f.p4,P & LE f.p4,f.p2,P or
          LE f.p3,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p1,P or
          LE f.p4,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p3,P or
          LE f.p2,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p4,P by JORDAN17:def 1;
          then f.p4=f.p2 or f.p2=f.p1 by A3,A9,A12,JGRAPH_3:26,JORDAN6:57;
          then
A37:      p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
          LE p2,p3,K by A36,Th50,JORDAN6:58;
          then p2=p3 by A6,A18,A30,A36,A37,JORDAN17:12,JORDAN6:57;
          hence contradiction by A6,A18,A30,A37,JORDAN17:12;
        end;
      end;
      hence contradiction;
    end;
    now per cases by A6,A14,A15,A16,A17,A18,JORDAN17:27;
      case p1,p2,p4,p3 are_in_this_order_on K;
        hence contradiction by A19;
      end;
      case
A38:    p1,p3,p2,p4 are_in_this_order_on K;
        now per cases by A38,JORDAN17:def 1;
          case
A39:        LE p1,p3,K & LE p3,p2,K & LE p2,p4,K;
            then f.p1,f.p3,f.p2,f.p4 are_in_this_order_on P by A1,A2,Th72;
            then LE f.p1,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p4,P or
            LE f.p3,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p1,P or
            LE f.p2,f.p4,P & LE f.p4,f.p1,P & LE f.p1,f.p3,P or
            LE f.p4,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p2,P
            by JORDAN17:def 1;
            then f.p3=f.p2 or LE f.p2,f.p1,P by A4,A13,JORDAN6:57,58;
            then f.p3=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
            then
A40:        p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
            then p3=p1 by A18,A38,A39,Th50,JORDAN6:57;
            hence contradiction by A18,A38,A40;
          end;
          case
A41:        LE p3,p2,K & LE p2,p4,K & LE p4,p1,K;
            then f.p3,f.p2,f.p4,f.p1 are_in_this_order_on P by A1,A2,Th72;
            then LE f.p1,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p4,P or
            LE f.p3,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p1,P or
            LE f.p2,f.p4,P & LE f.p4,f.p1,P & LE f.p1,f.p3,P or
            LE f.p4,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p2,P
            by JORDAN17:def 1;
            then f.p3=f.p2 or LE f.p2,f.p1,P by A4,A13,JORDAN6:57,58;
            then f.p3=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
            then
A42:        p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
            then p4=p1 by A18,A38,A41,Th50,JORDAN6:57;
            hence contradiction by A6,A18,A38,A42,JORDAN17:12;
          end;
          case
A43:        LE p2,p4,K & LE p4,p1,K & LE p1,p3,K;
            then f.p2,f.p4,f.p1,f.p3 are_in_this_order_on P by A1,A2,Th72;
            then LE f.p1,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p4,P or
            LE f.p3,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p1,P or
            LE f.p2,f.p4,P & LE f.p4,f.p1,P & LE f.p1,f.p3,P or
            LE f.p4,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p2,P
            by JORDAN17:def 1;
            then f.p3=f.p2 or LE f.p2,f.p1,P by A4,A13,JORDAN6:57,58;
            then f.p3=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
            then
A44:        p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
            then p4=p1 by A18,A38,A43,Th50,JORDAN6:57;
            hence contradiction by A6,A18,A38,A44,JORDAN17:12;
          end;
          case
A45:        LE p4,p1,K & LE p1,p3,K & LE p3,p2,K;
            then f.p4,f.p1,f.p3,f.p2 are_in_this_order_on P by A1,A2,Th72;
            then LE f.p1,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p4,P or
            LE f.p3,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p1,P or
            LE f.p2,f.p4,P & LE f.p4,f.p1,P & LE f.p1,f.p3,P or
            LE f.p4,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p2,P
            by JORDAN17:def 1;
            then f.p3=f.p2 or LE f.p2,f.p1,P by A4,A13,JORDAN6:57,58;
            then f.p3=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
            then
A46:        p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
            then p3=p1 by A18,A38,A45,Th50,JORDAN6:57;
            hence contradiction by A18,A38,A46;
          end;
        end;
        hence contradiction;
      end;
      case p1,p3,p4,p2 are_in_this_order_on K;
        hence contradiction by A29;
      end;
      case
A47:    p1,p4,p2,p3 are_in_this_order_on K;
        now per cases by A47,JORDAN17:def 1;
          case
A48:        LE p1,p4,K & LE p4,p2,K & LE p2,p3,K;
            then f.p1,f.p4,f.p2,f.p3 are_in_this_order_on P by A1,A2,Th72;
            then
A49:        LE f.p1,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p3,P or
            LE f.p4,f.p2,P & LE f.p2,f.p3,P & LE f.p3,f.p1,P or
            LE f.p2,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p4,P or
            LE f.p3,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p2,P
            by JORDAN17:def 1;
            then f.p4=f.p2 or LE f.p2,f.p1,P by A9,A13,JORDAN6:57,58;
            then f.p4=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
            then
A50:        p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
            then
A51:        p4=p2 by A48,Th50,JORDAN6:57;
            f.p3=f.p1 or LE f.p4,f.p3,P by A8,A13,A49,JORDAN6:57,58;
            then f.p3=f.p1 or f.p4=f.p3 by A5,A12,JGRAPH_3:26,JORDAN6:57;
            then
A52:        p3=p1 or p4=p3 by A2,A11,FUNCT_1:def 4;
            then p1=p2 by A18,A47,A48,A50,Th50,JORDAN6:57;
            hence contradiction by A18,A47,A51,A52;
          end;
          case
A53:        LE p4,p2,K & LE p2,p3,K & LE p3,p1,K;
            then f.p4,f.p2,f.p3,f.p1 are_in_this_order_on P by A1,A2,Th72;
            then
A54:        LE f.p1,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p3,P or
            LE f.p4,f.p2,P & LE f.p2,f.p3,P & LE f.p3,f.p1,P or
            LE f.p2,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p4,P or
            LE f.p3,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p2,P
            by JORDAN17:def 1;
            then f.p4=f.p2 or LE f.p2,f.p1,P by A9,A13,JORDAN6:57,58;
            then f.p4=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
            then p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
            then
A55:        p4=p2 or p2=p1 & p3=p1 by A53,Th50,JORDAN6:57;
            f.p3=f.p1 or LE f.p4,f.p3,P by A8,A13,A54,JORDAN6:57,58;
            then f.p3=f.p1 or f.p4=f.p3 by A5,A12,JGRAPH_3:26,JORDAN6:57;
            then p3=p1 or p4=p3 by A2,A11,FUNCT_1:def 4;
            hence contradiction by A6,A29,A47,A55,JORDAN17:12;
          end;
          case
A56:        LE p2,p3,K & LE p3,p1,K & LE p1,p4,K;
            then f.p2,f.p3,f.p1,f.p4 are_in_this_order_on P by A1,A2,Th72;
            then
A57:        LE f.p1,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p3,P or
            LE f.p4,f.p2,P & LE f.p2,f.p3,P & LE f.p3,f.p1,P or
            LE f.p2,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p4,P or
            LE f.p3,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p2,P
            by JORDAN17:def 1;
            then f.p4=f.p2 or LE f.p2,f.p1,P by A9,A13,JORDAN6:57,58;
            then f.p4=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
            then p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
            then
A58:        p4=p2 or p2=p1 & p3=p1 by A56,Th50,JORDAN6:57;
            f.p3=f.p1 or LE f.p4,f.p3,P by A8,A13,A57,JORDAN6:57,58;
            then f.p3=f.p1 or f.p4=f.p3 by A5,A12,JGRAPH_3:26,JORDAN6:57;
            then p3=p1 or p4=p3 by A2,A11,FUNCT_1:def 4;
            hence contradiction by A6,A29,A47,A58,JORDAN17:12;
          end;
          case
A59:        LE p3,p1,K & LE p1,p4,K & LE p4,p2,K;
            then f.p3,f.p1,f.p4,f.p2 are_in_this_order_on P by A1,A2,Th72;
            then
A60:        LE f.p1,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p3,P or
            LE f.p4,f.p2,P & LE f.p2,f.p3,P & LE f.p3,f.p1,P or
            LE f.p2,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p4,P or
            LE f.p3,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p2,P
            by JORDAN17:def 1;
            then f.p4=f.p2 or LE f.p2,f.p1,P by A9,A13,JORDAN6:57,58;
            then f.p4=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
            then p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
            then
A61:        p4=p2 by A59,Th50,JORDAN6:57;
            f.p3=f.p1 or LE f.p4,f.p3,P by A8,A13,A60,JORDAN6:57,58;
            then f.p3=f.p1 or f.p4=f.p3 by A5,A12,JGRAPH_3:26,JORDAN6:57;
            then p3=p1 or p4=p3 by A2,A11,FUNCT_1:def 4;
            hence contradiction by A6,A29,A47,A61,JORDAN17:12;
          end;
        end;
        hence contradiction;
      end;
      case
A62:    p1,p4,p3,p2 are_in_this_order_on K;
        now per cases by A62,JORDAN17:def 1;
          case
A63:        LE p1,p4,K & LE p4,p3,K & LE p3,p2,K;
            then f.p1,f.p4,f.p3,f.p2 are_in_this_order_on P by A1,A2,Th72;
            then
A64:        LE f.p1,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p2,P or
            LE f.p4,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p1,P or
            LE f.p3,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p4,P or
            LE f.p2,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p3,P
            by JORDAN17:def 1;
            then f.p3=f.p2 or f.p2=f.p1 by A3,A4,A12,JGRAPH_3:26,JORDAN6:57;
            then
A65:        p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
            LE p1,p3,K by A63,Th50,JORDAN6:58;
            then
A66:        p3=p2 by A63,A65,Th50,JORDAN6:57;
            f.p4=f.p3 or f.p2=f.p1 by A3,A5,A12,A64,JGRAPH_3:26,JORDAN6:57;
            then p4=p3 or p2=p1 by A2,A11,FUNCT_1:def 4;
            then p4=p3 or p2,p3,p4,p1 are_in_this_order_on K
            by A6,A62,A66,JORDAN17:12;
            hence contradiction by A6,A18,A62,A65,JORDAN17:12;
          end;
          case LE p4,p3,K & LE p3,p2,K & LE p2,p1,K;
            then f.p4,f.p3,f.p2,f.p1 are_in_this_order_on P by A1,A2,Th72;
            then
A67:        LE f.p1,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p2,P or
            LE f.p4,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p1,P or
            LE f.p3,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p4,P or
            LE f.p2,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p3,P
            by JORDAN17:def 1;
            then f.p3=f.p2 or f.p2=f.p1 by A3,A4,A12,JGRAPH_3:26,JORDAN6:57;
            then
A68:        p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
            f.p4=f.p3 or f.p2=f.p1 by A3,A5,A12,A67,JGRAPH_3:26,JORDAN6:57;
            then p4=p3 or p2=p1 by A2,A11,FUNCT_1:def 4;
            hence contradiction by A6,A19,A62,A68,JORDAN17:12;
          end;
          case LE p3,p2,K & LE p2,p1,K & LE p1,p4,K;
            then f.p3,f.p2,f.p1,f.p4 are_in_this_order_on P by A1,A2,Th72;
            then LE f.p1,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p2,P or
            LE f.p4,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p1,P or
            LE f.p3,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p4,P or
            LE f.p2,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p3,P
            by JORDAN17:def 1;
            then f.p4=f.p3 or f.p2=f.p1 by A3,A5,A12,JGRAPH_3:26,JORDAN6:57;
            then p4=p3 or p2=p1 by A2,A11,FUNCT_1:def 4;
            hence contradiction by A6,A19,A29,A62,JORDAN17:12;
          end;
          case
A69:        LE p2,p1,K & LE p1,p4,K & LE p4,p3,K;
            then f.p2,f.p1,f.p4,f.p3 are_in_this_order_on P by A1,A2,Th72;
            then
A70:        LE f.p1,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p2,P or
            LE f.p4,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p1,P or
            LE f.p3,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p4,P or
            LE f.p2,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p3,P
            by JORDAN17:def 1;
            then f.p3=f.p2 or f.p2=f.p1 by A3,A4,A12,JGRAPH_3:26,JORDAN6:57;
            then
A71:        p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
            LE p1,p3,K by A69,Th50,JORDAN6:58;
            then
A72:        p1=p2 by A69,A71,Th50,JORDAN6:57;
            f.p4=f.p3 or f.p2=f.p3 by A4,A5,A12,A70,JGRAPH_3:26,JORDAN6:57;
            then p4=p3 or p2=p3 by A2,A11,FUNCT_1:def 4;
            hence contradiction by A6,A29,A62,A72,JORDAN17:12;
          end;
        end;
        hence contradiction;
      end;
    end;
    hence contradiction;
  end;
  hence thesis;
end;
