
theorem :: Problem 153
  not ex x,y,z being positive Nat st
    x / y + y / z + z / x = 1
  proof
    given x,y,z being positive Nat such that
A1: x / y + y / z + z / x = 1;
DD: (x / y) * (y / z) * (z / x) = 1 by Multiply1;
    not (x / y < 1 & y / z < 1 & z / x < 1)
    proof
      assume
D1:   x / y < 1 & y / z < 1 & z / x < 1; then
      (x / y) * (y / z) < 1 by XREAL_1:162;
      hence thesis by DD,D1,XREAL_1:162;
    end; then
    per cases;
    suppose x / y >= 1; then
      x / y + y / z > 1 + 0 by XREAL_1:8;
      hence thesis by A1,XREAL_1:8;
    end;
    suppose y / z >= 1; then
      x / y + y / z > 1 + 0 by XREAL_1:8;
      hence thesis by A1,XREAL_1:8;
    end;
    suppose z / x >= 1; then
      z / x + y / z > 1 + 0 by XREAL_1:8; then
      x / y + (y / z + z / x) > 0 + 1 by XREAL_1:8;
      hence thesis by A1;
    end;
  end;
