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 Th7:
  for X being real-membered set holds X is non empty bounded_below
implies ex g st (for r st r in X holds g<=r) & for s st 0<s ex r st r in X & r<
  g+s
proof
  let X be real-membered set;
  assume that
A1: X is non empty and
A2: X is bounded_below;
A3: ex r1 being Real st r1 in X by A1;
  consider p1 such that
A4: p1 is LowerBound of X by A2;
A5: for r st r in X holds p1<=r by A4,XXREAL_2:def 2;
  reconsider X as Subset of REAL by MEMBERED:3;
  defpred X[Real] means for r st r in X holds $1<=r;
  consider Y such that
A6: for p be Element of REAL holds p in Y iff X[p] from SUBSET_1:sch 3;
  for p,r st p in Y & r in X holds p<=r by A6;
  then consider g1 such that
A7: for p,r st p in Y & r in X holds p<=g1 & g1<=r by AXIOMS:1;
  reconsider g1 as Real;
  take g=g1;
A8: now
    given s1 such that
A9: 0<s1 and
A10: for r st r in X holds not r<g+s1;
    reconsider gs1 = g+s1 as Element of REAL by XREAL_0:def 1;
    gs1 in Y by A6,A10;
    then g+s1<=g by A3,A7;
    then g+s1-g<=g-g by XREAL_1:9;
    hence contradiction by A9;
  end;
  p1 in REAL by XREAL_0:def 1;
  then p1 in Y by A5,A6;
  hence thesis by A7,A8;
end;
