
theorem Th16:
  for L being complete Lattice for a being Element of L holds a is
  completely-meet-irreducible implies a% is meet-irreducible
proof
  let L be complete Lattice;
  let a be Element of L;
  set X = {d where d is Element of L : a [= d & d <> a};
  assume a is completely-meet-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: a*' is_less_than X by LATTICE3:34;
    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 a% <= x by YELLOW_0:23;
    then
A9: a [= %x 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;
    a% <= y by A8,YELLOW_0:23;
    then
A13: a [= %y by A12,LATTICE3:7;
    y = %y by LATTICE3:def 4;
    then %y in X by A6,A13,A3,A11;
    then a*' [= %y by A5,LATTICE3:def 16;
    then
A14: (a*')% <= (%y)% by LATTICE3:7;
    x = %x by LATTICE3:def 4;
    then %x in X by A6,A9,A3,A10;
    then a*' [= %x by A5,LATTICE3:def 16;
    then (a*')% <= (%x)% by LATTICE3:7;
    then (a*')% <= a% by A8,A12,A7,A14,YELLOW_0:23;
    hence contradiction by A1,A2,A4,ORDERS_2:2;
  end;
  hence thesis by WAYBEL_6:def 2;
end;
