reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;

theorem Th6:
  for L1,L2 being Lattice st LattPOSet L1 = LattPOSet L2 holds
  the LattStr of L1 = the LattStr of L2
proof
  let L1,L2 be Lattice such that
A1: LattPOSet L1 = LattPOSet L2;
  reconsider j = join(L2), m = met(L2) as BinOp of carr(L1) by A1;
  now
    let a,b be Element of L1;
    reconsider x = a, y = b, xy = a"\/" b as Element of L2 by A1;
    reconsider ab = x"\/"y as Element of L1 by A1;
A2: a [= a"\/"b by LATTICES:5;
A3: b [= b"\/"a by LATTICES:5;
A4: x [= x"\/"y by LATTICES:5;
A5: y [= y"\/"x by LATTICES:5;
A6: [x,xy] in LattRel L2 by A1,A2,FILTER_1:31;
A7: [y,xy] in LattRel L2 by A1,A3,FILTER_1:31;
A8: [a,ab] in LattRel L1 by A1,A4,FILTER_1:31;
A9: [b,ab] in LattRel L1 by A1,A5,FILTER_1:31;
A10: a [= ab by A8,FILTER_1:31;
A11: b [= ab by A9,FILTER_1:31;
A12: x [= xy by A6,FILTER_1:31;
A13: y [= xy by A7,FILTER_1:31;
A14: a"\/"b [= ab by A10,A11,FILTER_0:6;
    x"\/"y [= xy by A12,A13,FILTER_0:6;
    then [ab,a"\/"b] in LattRel L1 by A1,FILTER_1:31;
    then ab [= a"\/"b by FILTER_1:31;
    hence join(L1).(a,b) = j.(a,b) by A14,LATTICES:8;
  end;
  then
A15: join(L1) = j by BINOP_1:2;
  now
    let a,b be Element of L1;
    reconsider x = a, y = b, xy = a"/\" b as Element of L2 by A1;
    reconsider ab = x"/\"y as Element of L1 by A1;
A16: a"/\"b [= a by LATTICES:6;
A17: b"/\"a [= b by LATTICES:6;
A18: x"/\"y [= x by LATTICES:6;
A19: y"/\"x [= y by LATTICES:6;
A20: [xy,x] in LattRel L2 by A1,A16,FILTER_1:31;
A21: [xy,y] in LattRel L2 by A1,A17,FILTER_1:31;
A22: [ab,a] in LattRel L1 by A1,A18,FILTER_1:31;
A23: [ab,b] in LattRel L1 by A1,A19,FILTER_1:31;
A24: ab [= a by A22,FILTER_1:31;
A25: ab [= b by A23,FILTER_1:31;
A26: xy [= x by A20,FILTER_1:31;
A27: xy [= y by A21,FILTER_1:31;
A28: ab [= a"/\"b by A24,A25,FILTER_0:7;
    xy [= x"/\"y by A26,A27,FILTER_0:7;
    then [a"/\"b,ab] in LattRel L1 by A1,FILTER_1:31;
    then a"/\"b [= ab by FILTER_1:31;
    hence met(L1).(a,b) = m.(a,b) by A28,LATTICES:8;
  end;
  hence thesis by A1,A15,BINOP_1:2;
end;
