
theorem Th8:
  for X,Y being non empty Subset of ExtREAL holds inf X + inf Y <= inf (X + Y)
proof
  let X,Y be non empty Subset of ExtREAL;
  for z being ExtReal holds z in X + Y implies inf X + inf Y <= z
  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: inf Y <= y by A3,XXREAL_2:3;
    inf X <= x by A2,XXREAL_2:3;
    hence thesis by A1,A4,XXREAL_3:36;
  end;
  then inf X + inf Y is LowerBound of X + Y by XXREAL_2:def 2;
  hence thesis by XXREAL_2:def 4;
end;
