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 Th45:
  for X being non empty real-membered set, r for t st for s st s
  in X holds s <= t holds upper_bound X <= t
proof
  let X be non empty real-membered set, r;
  set r = upper_bound X;
  let t;
  assume
A1: for s st s in X holds s <= t;
  set s = r-t;
  assume r > t; then
A2: s > 0 by XREAL_1:50;
  X is bounded_above proof take t;
    let s be ExtReal;
    thus thesis by A1;
   end;
  then ex t9 be Real st t9 in X & r-s < t9 by A2,Def1;
  hence contradiction by A1;
end;
