reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;

theorem
  X is real-bounded non empty implies ((ex r,p st r in X & p in X & p<>r) iff
  lower_bound X < upper_bound X)
proof
  assume that
A1: X is real-bounded and
A2: X is non empty;
  thus (ex r,p st r in X & p in X & p<>r) implies lower_bound X<upper_bound X
  proof
    given r,p such that
A3: r in X and
A4: p in X and
A5: p<>r;
A6: now
      assume
A7:   r<p;
      lower_bound X<=r by A1,A3,Def2;
      then
A8:   lower_bound X<p by A7,XXREAL_0:2;
      p<=upper_bound X by A1,A4,Def1;
      hence thesis by A8,XXREAL_0:2;
    end;
    now
      assume
A9:   p<r;
      lower_bound X<=p by A1,A4,Def2;
      then
A10:  lower_bound X<r by A9,XXREAL_0:2;
      r<=upper_bound X by A1,A3,Def1;
      hence thesis by A10,XXREAL_0:2;
    end;
    hence thesis by A5,A6,XXREAL_0:1;
  end;
  consider r being Element of REAL such that
A11: r in X by A2;
  assume that
A12: lower_bound X<upper_bound X and
A13: for r,p st r in X & p in X holds p=r;
  for x be object holds x in X iff x=r by A13,A11;
  then X={r} by TARSKI:def 1;
  hence contradiction by A12,Th10;
end;
