reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem
  for P being Subset of TOP-REAL 2, q1,q2,q3 being Point of TOP-REAL 2
  st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P holds LE q1,q3,P
proof
  let P be Subset of TOP-REAL 2, q1,q2,q3 be Point of TOP-REAL 2;
  assume that
A1: P is being_simple_closed_curve and
A2: LE q1,q2,P and
A3: LE q2,q3,P;
A4: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A1,Def9;
A5: Upper_Arc(P) /\ Lower_Arc(P)={W-min(P),E-max(P)} by A1,Def9;
A6: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by A1,Th50;
  now per cases by A2;
    case
A7:   q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P);
      now per cases by A3;
        case q2 in Upper_Arc(P) & q3 in Lower_Arc(P)& not q3=W-min(P);
          hence thesis by A7;
        end;
        case
A8:       q2 in Upper_Arc(P) & q3 in Upper_Arc(P) &
          LE q2,q3,Upper_Arc(P),W-min(P),E-max(P);
          then q2 in Upper_Arc(P) /\ Lower_Arc(P) by A7,XBOOLE_0:def 4;
          then q2=E-max(P) by A5,A7,TARSKI:def 2;
          hence thesis by A2,A6,A8,Th55;
        end;
        case q2 in Lower_Arc(P) & q3 in Lower_Arc(P)& not q3=W-min(P) &
          LE q2,q3,Lower_Arc(P),E-max(P),W-min(P);
          hence thesis by A7;
        end;
      end;
      hence thesis;
    end;
    case
A9:   q1 in Upper_Arc(P) & q2 in Upper_Arc(P)
      & LE q1,q2,Upper_Arc(P),W-min(P),E-max(P);
      now per cases by A3;
        case q2 in Upper_Arc(P) & q3 in Lower_Arc(P)& not q3=W-min(P);
          hence thesis by A9;
        end;
        case
      q2 in Upper_Arc(P) & q3 in Upper_Arc(P) &
          LE q2,q3,Upper_Arc(P),W-min(P),E-max(P);
          then LE q1,q3,Upper_Arc(P),W-min(P),E-max(P) by A9,JORDAN5C:13;
          hence thesis;
        end;
        case q2 in Lower_Arc(P) & q3 in Lower_Arc(P)& not q3=W-min(P) &
          LE q2,q3,Lower_Arc(P),E-max(P),W-min(P);
          hence thesis by A9;
        end;
      end;
      hence thesis;
    end;
    case
A10:  q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) &
      LE q1,q2,Lower_Arc(P),E-max(P),W-min(P);
      now per cases by A3;
        case
A11:      q2 in Upper_Arc(P) & q3 in Lower_Arc(P)& not q3=W-min(P);
          then q2 in Upper_Arc(P) /\ Lower_Arc(P) by A10,XBOOLE_0:def 4;
          then q2=E-max(P) by A5,A10,TARSKI:def 2;
          then LE q2,q3,Lower_Arc(P),E-max(P),W-min(P) by A4,A11,JORDAN5C:10;
          then LE q1,q3,Lower_Arc(P),E-max(P),W-min(P) by A10,JORDAN5C:13;
          hence thesis by A11;
        end;
        case
A12:      q2 in Upper_Arc(P) & q3 in Upper_Arc(P) &
          LE q2,q3,Upper_Arc(P),W-min(P),E-max(P);
          then q2 in Upper_Arc(P) /\ Lower_Arc(P) by A10,XBOOLE_0:def 4;
          then q2=E-max(P) by A5,A10,TARSKI:def 2;
          hence thesis by A2,A6,A12,Th55;
        end;
        case
A13:      q2 in Lower_Arc(P) & q3 in Lower_Arc(P)& not q3=W-min(P) &
          LE q2,q3,Lower_Arc(P),E-max(P),W-min(P);
          then LE q1,q3,Lower_Arc(P),E-max(P),W-min(P) by A10,JORDAN5C:13;
          hence thesis by A13;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
