reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th5:
  for r,s be Real holds max+(r+s) <= max+(r) + max+(s)
proof
  let r,s be Real;
A1: 0<=max(r,0) & 0<=max(s,0) by XXREAL_0:25;
A2: r<=max(r,0) & s<=max(s,0) by XXREAL_0:25;
  now
    per cases;
    case
      0<=r+s;
      then max+(r+s) = r+s by XXREAL_0:def 10;
      hence thesis by A2,XREAL_1:7;
    end;
    case
      r+s<0;
      then max+(r+s) = 0+0 by XXREAL_0:def 10;
      hence thesis by A1;
    end;
  end;
  hence thesis;
end;
