reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;

theorem Th4:
  for a,c being Real, b being R_eal st a < b & [.a,b.[ c= [.a,c.[ holds
  b is Real
  proof
    let a,c be Real, b be R_eal;
    assume that
A1: a < b and
A2: [.a,b.[ c= [.a,c.[;
    set K = [.a,b.[;
A3: not c in K by A2,XXREAL_1:3;
    per cases;
    suppose
A4:   K is non empty;
      then consider x be object such that
A5:   x in K;
      reconsider x as Real by A5;
      assume not b is Real;
      then not b in REAL & a in REAL & a <= b by XREAL_0:def 1,A4,XXREAL_1:27;
      then
A6:   b = +infty by XXREAL_0:10;
      a <= x by A5,XXREAL_1:3; then
A7:   [.x,+infty.[ c= [.a,b.[ by  A6,XXREAL_1:38;
      per cases;
      suppose c < x;
        hence contradiction by A5,A2,XXREAL_1:3;
      end;
      suppose x <= c;
        hence contradiction by A3,A7,XXREAL_1:236;
      end;
    end;
    suppose K is empty;
      hence thesis by A1,XXREAL_1:31;
    end;
  end;
