reserve a,b,c,x,y,z for Real;

theorem
  for x,y being Real st 0 <= x & 0 <= y holds max(x,y) <= x + y
proof
  let x,y be Real;
  assume that
A1: 0 <= x and
A2: 0 <= y;
  now
    per cases by XXREAL_0:16;
    suppose
A3:   max(x,y) = x;
      x + 0 <= x + y by A2,XREAL_1:7;
      hence thesis by A3;
    end;
    suppose
A4:   max(x,y) = y;
      y + 0 <= y + x by A1,XREAL_1:7;
      hence thesis by A4;
    end;
  end;
  hence thesis;
end;
