
theorem Th18:
  for L being complete Lattice for a being Element of L holds a is
  completely-join-irreducible implies a% is join-irreducible
proof
  let L be complete Lattice;
  let a be Element of L;
  set X = {d where d is Element of L : d [= a & d <> a};
  assume a is completely-join-irreducible;
  then
A1: a <> *'a;
  for x,y being Element of LattPOSet L st a% = x "\/" y holds x = a% or y = a%
  proof
    *'a [= a by Th9;
    then
A2: a% >= (*'a)% by LATTICE3:7;
A3: %(a%) = a% by LATTICE3:def 4;
A4: a = a% & *'a = (*'a)% by LATTICE3:def 3;
A5: X is_less_than *'a by LATTICE3:def 21;
    let x,y be Element of LattPOSet L;
A6: a = a% by LATTICE3:def 3
      .= %(a%) by LATTICE3:def 4;
A7: x = %x by LATTICE3:def 4
      .= (%x)% by LATTICE3:def 3;
    assume
A8: a% = x "\/" y;
    then x <= a% by YELLOW_0:22;
    then
A9: %x [= a by A7,LATTICE3:7;
    assume that
A10: x <> a% and
A11: y <> a%;
A12: y = %y by LATTICE3:def 4
      .= (%y)% by LATTICE3:def 3;
    y <= a% by A8,YELLOW_0:22;
    then
A13: %y [= a by A12,LATTICE3:7;
    y = %y by LATTICE3:def 4;
    then %y in X by A6,A13,A3,A11;
    then %y [= *'a by A5,LATTICE3:def 17;
    then
A14: (*'a)% >= y by A12,LATTICE3:7;
    x = %x by LATTICE3:def 4;
    then %x in X by A6,A9,A3,A10;
    then %x [= *'a by A5,LATTICE3:def 17;
    then (*'a)% >= x by A7,LATTICE3:7;
    then (*'a)% >= a% by A8,A14,YELLOW_0:22;
    hence contradiction by A1,A2,A4,ORDERS_2:2;
  end;
  hence thesis by WAYBEL_6:def 3;
end;
