
theorem Th1:
  for x, y be Nat st x > 1 & y > 1 holds x*y >= x+y
  proof
    let x, y be Nat;
    assume
A1: x > 1 & y > 1;
    per cases;
    suppose
A2:   x >= y;
      y - 0 > 1 by A1; then
      y - 1 >= 0 + 1 by INT_1:7,XREAL_1:12; then
      x * (y - 1) >= y * 1 by A2,XREAL_1:66; then
      x * y - x + x >= y + x by XREAL_1:6;
      hence thesis;
    end;
    suppose
A3:   x < y;
      x - 0 > 1 by A1; then
      x - 1 >= 0 + 1 by XREAL_1:12,INT_1:7; then
      y * (x - 1) >= x * 1 by A3,XREAL_1:66; then
      x * y - y + y >= x + y by XREAL_1:6;
      hence thesis;
    end;
  end;
