
theorem Th3:
  for L being noetherian Lattice for a being Element of L for d
being Element of L st a [= d & a <> d holds ex c being Element of L st c [= d &
  c is-upper-neighbour-of a
proof
  let L be noetherian Lattice;
  let a be Element of L;
  let d be Element of L;
  defpred P[Element of LattPOSet L] means a [= %($1) & a <> %($1) implies ex c
  being Element of L st c [= %($1) & c is-upper-neighbour-of a;
A1: %(d%) = d% by LATTICE3:def 4
    .= d by LATTICE3:def 3;
A2: for x being Element of LattPOSet L st for y being Element of LattPOSet L
  st y <> x & [y,x] in the InternalRel of LattPOSet L holds P[y] holds P[x]
  proof
    let x be Element of LattPOSet L;
    assume
A3: for y being Element of LattPOSet L st y <> x & [y,x] in the
    InternalRel of LattPOSet L holds P[y];
    a [= %x & a <> %x implies ex c being Element of L st c [= %x & c
    is-upper-neighbour-of a
    proof
A4:   (%x)% = %x by LATTICE3:def 3
        .= x by LATTICE3:def 4;
      assume
A5:   a [= %x & a <> %x;
      per cases;
      suppose
        %x is-upper-neighbour-of a;
        hence thesis;
      end;
      suppose
        not %x is-upper-neighbour-of a;
        then consider c being Element of L such that
A6:     a [= c and
A7:     c [= %x and
A8:     ( not c = %x)& not c = a by A5;
        c% <= x by A4,A7,LATTICE3:7;
        then
A9:     [c%,x] in the InternalRel of LattPOSet L by ORDERS_2:def 5;
        %(c%) = c% by LATTICE3:def 4
          .= c by LATTICE3:def 3;
        then ex c9 being Element of L st c9 [= c & c9 is-upper-neighbour-of a
        by A3,A6,A8,A9;
        hence thesis by A7,LATTICES:7;
      end;
    end;
    hence thesis;
  end;
A10: LattPOSet L is well_founded by Def3;
A11: for x being Element of LattPOSet L holds P[x] from WELLFND1:sch 3(A2,
  A10);
  assume a [= d & a <> d;
  hence thesis by A11,A1;
end;
