theorem
  for X,Y being real-membered set st X<>{} & Y<>{} & for a,b st a in X &
b in Y holds a<=b holds ex d st (for a st a in X holds a<=d) & for b st b in Y
  holds d<=b
proof
  let X,Y be real-membered set;
  set x = the Element of X;
  reconsider a=x as Real;
  set y = the Element of Y;
  reconsider b=y as Real;
  assume X<>{} & Y<>{};
  then
A1: a in X & b in Y;
A2: X c= REAL & Y c= REAL by Th3;
  assume for a,b st a in X & b in Y holds a<=b;
  then consider d being Real such that
A3: for a,b being Real st a in X & b in Y holds a<=d & d<=b by A2,
AXIOMS:1;
  reconsider d as Real;
  take d;
  thus thesis by A1,A3;
end;
