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

theorem Th65:
  for X,Y being ext-real-membered set, x being UpperBound of X, y
  being UpperBound of Y holds min(x,y) is UpperBound of X /\ Y
proof
  let X,Y be ext-real-membered set, x be UpperBound of X, y be UpperBound of Y;
  let a be ExtReal;
  assume
A1: a in X /\ Y;
  then a in Y by XBOOLE_0:def 4;
  then
A2: a <= y by Def1;
  a in X by A1,XBOOLE_0:def 4;
  then a <= x by Def1;
  hence thesis by A2,XXREAL_0:20;
end;
