reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;

theorem Th66:
  for X,Y being ext-real-membered set, x being LowerBound of X, y
  being LowerBound of Y holds max(x,y) is LowerBound of X /\ Y
proof
  let X,Y be ext-real-membered set, x be LowerBound of X, y be LowerBound of Y;
  let a be ExtReal;
  assume
A1: a in X /\ Y;
  then a in Y by XBOOLE_0:def 4;
  then
A2: y <= a by Def2;
  a in X by A1,XBOOLE_0:def 4;
  then x <= a by Def2;
  hence thesis by A2,XXREAL_0:28;
end;
