reserve C, P for Simple_closed_curve,
  a, b, c, d, e for Point of TOP-REAL 2;

theorem Th6:
  for P being Subset of TOP-REAL 2 st a <> b & P is_an_arc_of c,d &
  LE a,b,P,c,d holds ex e st a <> e & b <> e & LE a,e,P,c,d & LE e,b,P,c,d
proof
  let P be Subset of TOP-REAL 2;
  assume that
A1: a <> b and
A2: P is_an_arc_of c,d and
A3: LE a,b,P,c,d;
  b in P by A3,JORDAN5C:def 3;
  then consider
  f being Function of I[01], (TOP-REAL 2)|P, rb being Real such that
A4: f is being_homeomorphism and
A5: f.0 = c & f.1 = d and
A6: 0 <= rb and
A7: rb <= 1 and
A8: f.rb = b by A2,Th1;
A9: rng f = [#]((TOP-REAL 2)|P) by A4,TOPS_2:def 5
    .= the carrier of (TOP-REAL 2)|P
    .= P by PRE_TOPC:8;
  a in P by A3,JORDAN5C:def 3;
  then consider ra being object such that
A10: ra in dom f and
A11: a = f.ra by A9,FUNCT_1:def 3;
A12: dom f = [#]I[01] by A4,TOPS_2:def 5
    .= [.0,1.] by BORSUK_1:40;
  then reconsider ra as Real by A10;
A13: 0 <= ra by A10,A12,XXREAL_1:1;
A14: ra <= 1 by A10,A12,XXREAL_1:1;
  then ra <= rb by A3,A4,A5,A6,A7,A8,A11,A13,JORDAN5C:def 3;
  then ra < rb by A1,A8,A11,XXREAL_0:1;
  then consider re being Real such that
A15: ra < re and
A16: re < rb by XREAL_1:5;
  set e = f.re;
A17: re <= 1 by A7,A16,XXREAL_0:2;
A18: 0 <= re by A13,A15,XXREAL_0:2;
  then
A19: re in dom f by A12,A17,XXREAL_1:1;
  then e in rng f by FUNCT_1:def 3;
  then reconsider e as Point of TOP-REAL 2 by A9;
  take e;
  now
    assume
A20: a = e or b = e;
    f is one-to-one & rb in dom f by A4,A6,A7,A12,TOPS_2:def 5,XXREAL_1:1;
    hence contradiction by A8,A10,A11,A15,A16,A19,A20,FUNCT_1:def 4;
  end;
  hence a <> e & b <> e;
  thus thesis by A2,A4,A5,A6,A7,A8,A11,A13,A14,A15,A16,A18,A17,JORDAN5C:8;
end;
