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 Th43:
  for X being non empty real-membered set for t st for s st s in X
  holds s >= t holds lower_bound X >= t
proof
  let X be non empty real-membered set;
  set r = lower_bound X;
  let t;
  assume
A1: for s st s in X holds s >= t;
  set s = t-r;
  assume r < t; then
A2: s > 0 by XREAL_1:50;
  X is bounded_below proof take t; let s be ExtReal;
    thus thesis by A1;
   end;
  then ex t9 be Real st t9 in X & t9 < r+s by A2,Def2;
  hence contradiction by A1;
end;
