 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 Th6:
  for L1,L2 being WA_Lattice st LatRelStr L1 = LatRelStr L2 holds
    the LattStr of L1 = the LattStr of L2
  proof
    let L1,L2 be WA_Lattice such that
A1: LatRelStr L1 = LatRelStr L2;
    reconsider j = the L_join of L2, m = the L_meet of L2
      as BinOp of the carrier of 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;
      a [= a "\/" b by LATWAL_1:9; then
A6:   [x,xy] in LatOrder L2 by A1;
      b [= b "\/" a by LATWAL_1:9; then
A7:   [y,xy] in LatOrder L2 by A1;
      x [= x "\/" y by LATWAL_1:9; then
A8:   [a,ab] in LatOrder L1 by A1;
      y [= y "\/" x by LATWAL_1:9; then
A9:   [b,ab] in LatOrder L1 by A1;
A10:  a [= ab by A8,Idem2;
A11:  b [= ab by A9,Idem2;
A12:  x [= xy by A6,Idem2;
A13:  y [= xy by A7,Idem2;
A14:  a "\/" b [= ab by A10,A11,LATWAL_1:11;
      x "\/" y [= xy by A12,A13,LATWAL_1:11;
      then [ab,a "\/" b] in LatOrder L1 by A1;
      then ab [= a "\/" b by Idem2;
      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;
      x "/\" y [= x by LATTICES:6; then
      [ab,a] in LatOrder L1 by A1; then
A24:  ab [= a by Idem2;
      y "/\" x [= y by LATTICES:6; then
      [ab,b] in LatOrder L1 by A1; then
A25:  ab [= b by Idem2;
      a "/\" b [= a by LATTICES:6; then
      [xy,x] in LatOrder L2 by A1; then
A26:  xy [= x by Idem2;
      b "/\" a [= b by LATTICES:6; then
      [xy,y] in LatOrder L2 by A1; then
A27:  xy [= y by Idem2;
A28:  ab [= a "/\" b by A24,A25,LATWAL_1:13;
      xy [= x "/\" y by A26,A27,LATWAL_1:13;
      then [a "/\" b,ab] in LatOrder L1 by A1;
      then a "/\" b [= ab by Idem2;
      hence met(L1).(a,b) = m.(a,b) by A28,LATTICES:8;
    end;
    hence thesis by A1,A15,BINOP_1:2;
  end;
