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;
