 reserve W for WA-Lattice;
 reserve a,b,c for Element of W;
 reserve W for pcs-Compatible pcs-tol-reflexive pcs-tol-symmetric WAP-Lattice;
 reserve a,b for Element of W;
 reserve L for WA_Lattice;

theorem
  for L being WA-Lattice st
    the InternalRel of L is transitive holds
      wlatt L is Lattice
  proof
    let L be WA-Lattice;
A1: field the InternalRel of L = the carrier of L by ORDERS_1:12;
    assume the InternalRel of L is transitive; then
    the InternalRel of L is_transitive_in field the InternalRel of L
      by RELAT_2:def 16; then
    reconsider LL = L as with_suprema with_infima Poset
      by ORDERS_2:def 3,A1;
B1: the RelStr of LL = LatRelStr wlatt LL by WLatDef;
BC: the RelStr of LL = LattPOSet latt LL by LATTICE3:def 15; then
BB: LatRelStr wlatt LL = LattPOSet latt LL by B1;
    set L1 = latt LL;
    set L2 = wlatt LL;
D2: the carrier of L1 = the carrier of L2 by BB;
    LatOrder latt LL = LatOrder wlatt LL
    proof
T1:   LatOrder latt LL c= LatOrder wlatt LL
      proof
        let x,y be object;
        assume [x,y] in LatOrder latt LL; then
        consider a,b being Element of latt LL such that
A1:     [x,y] = [a,b] & a [= b;
        reconsider aa = a, bb = b as Element of wlatt LL by D2;
        reconsider a1 = a, b1 = b as Element of L by BC;
        a1 <= b1 by A1,Eq1; then
        [aa,bb] = [x,y] & aa [= bb by A1,Eq2;
        hence thesis;
      end;
      LatOrder wlatt LL c= LatOrder latt LL
      proof
        let x,y be object;
        assume [x,y] in LatOrder wlatt LL; then
        consider a,b being Element of wlatt LL such that
A1:     [x,y] = [a,b] & a [= b;
        reconsider aa = a, bb = b as Element of latt LL by D2;
        reconsider a1 = a, b1 = b as Element of L by B1;
        a1 <= b1 by A1,Eq2; then
        [aa,bb] = [x,y] & aa [= bb by A1,Eq1;
        hence thesis;
      end;
      hence thesis by T1,XBOOLE_0:def 10;
    end; then
    LatRelStr L1 = LatRelStr L2 by D2;
    hence thesis by Th6;
  end;
