
theorem Th4:
for x,y,z being Integer holds max(x+y,x+z) = x + max(y,z)
proof
let x,y,z be Integer;
per cases;
suppose A1: y <= z;
  then x+y <= x+z by XREAL_1:6;
  hence max(x+y,x+z) = x+z by XXREAL_0:def 10
                    .= x + max(y,z) by A1,XXREAL_0:def 10;
  end;
suppose A2: y > z;
  then x+y > x+z by XREAL_1:8;
  hence max(x+y,x+z) = x+y by XXREAL_0:def 10
                    .= x + max(y,z) by A2,XXREAL_0:def 10;
  end;
end;
