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 Th1:
  for X,Y st for r,p st r in X & p in Y holds r<p ex g st for r,p
  st r in X & p in Y holds r<=g & g<=p
proof
  let X,Y;
  assume for r,p st r in X & p in Y holds r<p;
  then for r,p st r in X & p in Y holds r<=p;
  then consider g such that
A1: for r,p st r in X & p in Y holds r <= g & g <= p by AXIOMS:1;
  reconsider g as Real;
  take g;
  thus thesis by A1;
end;
