reserve n for non zero Nat,
  j,k,l,m for Nat,
  g,h,i for Integer;

theorem Th9:
  g <= h + i & h < 0 & i < 0 implies g < h & g < i
proof
  assume that
A1: g <= h + i and
A2: h < 0 and
A3: i < 0;
  g - i <= h + i - i by A1,XREAL_1:9;
  then i + (g + -i) < 0 + h by A3,XREAL_1:8;
  hence g < h;
  g - h <= i + h - h by A1,XREAL_1:9;
  then h + (g + -h) < 0 + i by A2,XREAL_1:8;
  hence thesis;
end;
