
theorem Th12:
  for L being complete Lattice for a being Element of L st a is
  completely-meet-irreducible holds a*' is-upper-neighbour-of a & for c being
  Element of L holds c is-upper-neighbour-of a implies c = a*'
proof
  let L be complete Lattice;
  let a be Element of L;
  set X = { x where x is Element of L : a [= x & x <> a};
  for c being Element of L st a [= c & c [= a*' holds c = a or c = a*'
  proof
    let c be Element of L;
    assume that
A1: a [= c and
A2: c [= a*';
    assume c <> a;
    then c in X by A1;
    then a*' [= c by LATTICE3:38;
    hence thesis by A2,LATTICES:8;
  end;
  then
A3: for c being Element of L holds a [= c & c [= a*' implies (c = a*' or c
  = a);
  assume a is completely-meet-irreducible;
  then
A4: a*' <> a;
A5: for c being Element of L holds c is-upper-neighbour-of a implies c = a *'
  proof
    let c be Element of L;
    assume
A6: c is-upper-neighbour-of a;
    then a <> c & a [= c;
    then c in X;
    then
A7: a*' [= c by LATTICE3:38;
    a [= a*' by Th9;
    hence thesis by A4,A6,A7;
  end;
  a [= a*' by Th9;
  hence thesis by A4,A3,A5;
end;
