
theorem Th7:
  for X,Y being non empty Subset of ExtREAL holds sup (X + Y) <= sup X + sup Y
proof
  let X,Y be non empty Subset of ExtREAL;
  for z being ExtReal holds z in X + Y implies z <= sup X + sup Y
  proof
    let z be ExtReal;
    assume z in X + Y;
    then consider x,y being R_eal such that
A1: z = x + y and
A2: x in X and
A3: y in Y;
A4: y <= sup Y by A3,XXREAL_2:4;
    x <= sup X by A2,XXREAL_2:4;
    hence thesis by A1,A4,XXREAL_3:36;
  end;
  then sup X + sup Y is UpperBound of X + Y by XXREAL_2:def 1;
  hence thesis by XXREAL_2:def 3;
end;
