reserve C for Simple_closed_curve,
  p,q,p1 for Point of TOP-REAL 2,
  i,j,k,n for Nat,
  r,s for Real;

theorem Th10:
  LE E-max C, q, C implies
  Segment(q,W-min C,C) = Segment(Lower_Arc C,E-max C,W-min C,q,W-min C)
proof
  set p = W-min C;
  assume
A1: LE E-max C, q, C;
A2: Lower_Arc C is_an_arc_of E-max C,W-min C by JORDAN6:50;
A3: p in Lower_Arc C by JORDAN7:1;
A4: q in Lower_Arc C by A1,JORDAN17:4;
A5: Lower_Arc C c= C by JORDAN6:61;
  defpred P[Point of TOP-REAL 2] means LE q,$1,C or q in C & $1=W-min C;
  defpred Q[Point of TOP-REAL 2] means LE q,$1,Lower_Arc C,E-max C,W-min C &
  LE $1,p,Lower_Arc C,E-max C, W-min C;
A6: P[p1] iff Q[p1]
  proof
    hereby
      assume
A7:   LE q,p1,C or q in C & p1=W-min C;
      per cases by A7;
      suppose that
A8:     q = E-max C and
A9:     LE q,p1,C;
A10:    p1 in Lower_Arc C by A8,A9,JORDAN17:4;
        hence LE q,p1,Lower_Arc C,E-max C,W-min C by A2,A8,JORDAN5C:10;
        thus LE p1,p,Lower_Arc C,E-max C,W-min C by A2,A10,JORDAN5C:10;
      end;
      suppose that
A11:    q <> E-max C and
A12:    LE q,p1,C;
A13:    p1 in Lower_Arc C by A1,A12,JORDAN17:4,JORDAN6:58;
A14:    now
          assume
A15:      q = W-min C;
          then LE q, E-max C, C by JORDAN7:3,SPRECT_1:14;
          hence contradiction by A1,A15,JORDAN6:57,TOPREAL5:19;
        end;
        now
          assume q in Upper_Arc C;
          then q in Upper_Arc C /\ Lower_Arc C by A4,XBOOLE_0:def 4;
          then q in {E-max C, W-min C} by JORDAN6:def 9;
          hence contradiction by A11,A14,TARSKI:def 2;
        end;
        hence LE q,p1,Lower_Arc C,E-max C,W-min C by A12,JORDAN6:def 10;
        thus LE p1,p,Lower_Arc C,E-max C,W-min C by A2,A13,JORDAN5C:10;
      end;
      suppose that q in C and
A16:    p1 = W-min C;
        thus LE q,p1,Lower_Arc C,E-max C,W-min C by A2,A4,A16,JORDAN5C:10;
        thus LE p1,p,Lower_Arc C,E-max C,W-min C by A3,A16,JORDAN5C:9;
      end;
    end;
    assume that
A17: LE q,p1,Lower_Arc C,E-max C,W-min C and
    LE p1,p,Lower_Arc C,E-max C,W-min C;
A18: p1 in Lower_Arc C by A17,JORDAN5C:def 3;
A19: q in Lower_Arc C by A17,JORDAN5C:def 3;
    per cases;
    suppose p1 <> W-min C;
      hence thesis by A17,A18,A19,JORDAN6:def 10;
    end;
    suppose p1 = W-min C;
      hence thesis by A4,A5;
    end;
  end;
  deffunc F(set) = $1;
  set X = {F(p1): P[p1]}, Y = {F(p1): Q[p1]};
A20: X = Y from FRAENKEL:sch 3(A6);
  Segment(q,p,C) = X by JORDAN7:def 1;
  hence thesis by A20,JORDAN6:26;
end;
