
theorem Th4:
  for L being co-noetherian Lattice for a being Element of L for d
being Element of L st d [= a & a <> d holds ex c being Element of L st d [= c &
  c is-lower-neighbour-of a
proof
  let L be co-noetherian Lattice;
  let a be Element of L;
  let d be Element of L;
  defpred P[Element of (LattPOSet L)~] means %(~($1)) [= a & a <> %(~($1))
  implies ex c being Element of L st %(~($1)) [= c & c is-lower-neighbour-of a;
A1: %(~((d%)~)) = ~((d%)~) by LATTICE3:def 4
    .= (d%)~ by LATTICE3:def 7
    .= d% by LATTICE3:def 6
    .= 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];
    %(~x) [= a & a <> %(~x) implies ex c being Element of L st %(~x) [= c
    & c is-lower-neighbour-of a
    proof
A4:   (~x)~ = ~x by LATTICE3:def 6
        .= x by LATTICE3:def 7;
A5:   (%(~x))% = %(~x) by LATTICE3:def 3
        .= ~x by LATTICE3:def 4;
      assume
A6:   %(~x) [= a & a <> %(~x);
      per cases;
      suppose
        %(~x) is-lower-neighbour-of a;
        hence thesis;
      end;
      suppose
        not %(~x) is-lower-neighbour-of a;
        then consider c being Element of L such that
A7:     %(~x) [= c and
A8:     c [= a & not c = a & not c = %(~x) by A6;
A9:     %(~((c%)~)) = ~((c%)~) by LATTICE3:def 4
          .= (c%)~ by LATTICE3:def 7
          .= c% by LATTICE3:def 6
          .= c by LATTICE3:def 3;
        ~x <= c% by A5,A7,LATTICE3:7;
        then (c%)~ <= x by A4,LATTICE3:9;
        then [(c%)~,x] in the InternalRel of (LattPOSet L)~ by ORDERS_2:def 5;
        then ex c9 being Element of L st %(~((c%)~)) [= c9 & c9
        is-lower-neighbour-of a by A3,A8,A9;
        hence thesis by A7,A9,LATTICES:7;
      end;
    end;
    hence thesis;
  end;
A10: (LattPOSet L)~ is well_founded by Def4;
A11: for x being Element of (LattPOSet L)~ holds P[x] from WELLFND1:sch 3 (
  A2,A10);
  assume d [= a & a <> d;
  hence thesis by A11,A1;
end;
