reserve n,k for Element of NAT;

theorem Th15:
  for L be finite LATTICE holds LOWER(subrelstr Join-IRR L) is Ring_of_sets
proof
  let L be finite LATTICE;
  set X = LOWER(subrelstr Join-IRR L);
  X includes_lattice_of X
  proof
    let a,b be set;
    assume that
A1: a in X and
A2: b in X;
A3: a \/ b in X
    proof
      consider Z1 be Subset of subrelstr Join-IRR L such that
A4:   b=Z1 and
A5:   Z1 is lower by A2;
      consider Z be Subset of subrelstr Join-IRR L such that
A6:   a=Z and
A7:   Z is lower by A1;
      Z \/ Z1 is lower by A7,A5,WAYBEL_0:27;
      hence thesis by A6,A4;
    end;
    a /\ b in X
    proof
      consider Z1 be Subset of subrelstr Join-IRR L such that
A8:   b=Z1 and
A9:   Z1 is lower by A2;
      consider Z be Subset of subrelstr Join-IRR L such that
A10:  a=Z and
A11:  Z is lower by A1;
      Z /\ Z1 is lower by A11,A9,WAYBEL_0:27;
      hence thesis by A10,A8;
    end;
    hence thesis by A3;
  end;
  hence thesis by Def8;
end;
