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 Th8:
  for X being real-membered set holds (for r st r in X holds g1<=r
) & (for s st 0<s ex r st (r in X & r<g1+s)) & (for r st r in X holds g2<=r) &
  (for s st 0<s ex r st (r in X & r<g2+s)) implies g1=g2
proof
  let X be real-membered set;
  assume that
A1: for r st r in X holds g1<=r and
A2: for s st 0<s ex r st r in X & r<g1+s and
A3: for r st r in X holds g2<=r and
A4: for s st 0<s ex r st r in X & r<g2+s;
A5: now
    assume g2<g1;
    then ex r1 st r1 in X & r1<g2+(g1-g2) by A4,XREAL_1:50;
    hence contradiction by A1;
  end;
  now
    assume g1<g2;
    then ex r1 st r1 in X & r1<g1+(g2-g1) by A2,XREAL_1:50;
    hence contradiction by A3;
  end;
  hence thesis by A5,XXREAL_0:1;
end;
