
theorem Th14:
  for R being with_suprema non empty Poset, l being Element of R
  holds l is co-prime iff for x,y be Element of R st l <= x "\/" y holds l <= x
  or l <= y
proof
  let R be with_suprema non empty Poset, l be Element of R;
  hereby
    assume l is co-prime;
    then
A1: l~ is prime;
    let x, y be Element of R;
    assume l <= x "\/" y;
    then
A2: (x "\/" y)~ <= l~ by LATTICE3:9;
    (x "\/" y)~ = x"\/"y by LATTICE3:def 6
      .= (x~)"/\"(y~) by YELLOW_7:23;
    then x~ <= l~ or y~ <= l~ by A1,A2;
    hence l <= x or l <= y by LATTICE3:9;
  end;
  assume
A3: for x,y be Element of R st l <= x "\/" y holds l <= x or l <= y;
  let x,y be Element of R~;
A4: ~(x "/\" y) = x "/\" y by LATTICE3:def 7
    .= ~x"\/"~y by YELLOW_7:24;
  assume x "/\" y <= l~;
  then l <= ~x"\/"~y by A4,YELLOW_7:2;
  then l <= ~x or l <= ~y by A3;
  hence x <= l~ or y <= l~ by YELLOW_7:2;
end;
