reserve Y for non empty set,
  a for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  P,Q for a_partition of Y;

theorem Th15:
  G is independent implies All(All(a,P,G),Q,G) = All(All(a,Q,G),P,G)
proof
  set A = G \ {P}, B = G \ {Q};
A1: CompF(P,G) = '/\' A by BVFUNC_2:def 7;
A2: A c= G & B c= G by XBOOLE_1:36;
A3: CompF(Q,G) = '/\' B by BVFUNC_2:def 7;
  assume G is independent;
  then
A4: ERl('/\'A)*ERl '/\'B = ERl ('/\'B)*ERl '/\'A by A2,Th14;
A5: for y being Element of Y holds ( (for x being Element of Y st x in
EqClass(y,CompF(Q,G)) holds B_INF(a,CompF(P,G)).x=TRUE) implies B_INF( B_INF(a,
  CompF(Q,G)),CompF(P,G)).y = TRUE ) & (not (for x being Element of Y st x in
EqClass(y,CompF(Q,G)) holds B_INF(a,CompF(P,G)).x=TRUE) implies B_INF( B_INF(a,
  CompF(Q,G)),CompF(P,G)).y = FALSE)
  proof
    let y be Element of Y;
    hereby
      assume
A6:   for x being Element of Y st x in EqClass(y,CompF(Q,G)) holds
      B_INF(a,CompF(P,G)).x=TRUE;
      for x being Element of Y st x in EqClass(y,CompF(P,G)) holds B_INF(a
      ,CompF(Q,G)).x=TRUE
      proof
        let x be Element of Y;
        assume x in EqClass(y,CompF(P,G)); then
A7:     [x,y] in ERl '/\' A by A1,Th5;
        for z being Element of Y st z in EqClass(x,CompF(Q,G)) holds a.z= TRUE
        proof
          let z be Element of Y;
          assume z in EqClass(x,CompF(Q,G));
          then [z,x] in ERl '/\' B by A3,Th5;
          then [z,y] in (ERl '/\' A)*ERl '/\' B by A4,A7,RELAT_1:def 8;
          then consider u being object such that
A8:       [z,u] in ERl '/\' A and
A9:       [u,y] in ERl '/\' B by RELAT_1:def 8;
          u in field ERl '/\' B by A9,RELAT_1:15;
          then reconsider u as Element of Y by ORDERS_1:12;
          u in EqClass(y,CompF(Q,G)) by A3,A9,Th5;
          then
A10:      B_INF(a,CompF(P,G)).u <> FALSE by A6;
          z in EqClass(u,CompF(P,G)) by A1,A8,Th5;
          hence thesis by A10,BVFUNC_1:def 16;
        end;
        hence thesis by BVFUNC_1:def 16;
      end;
      hence B_INF(B_INF(a,CompF(Q,G)),CompF(P,G)).y = TRUE by BVFUNC_1:def 16;
    end;
    given x being Element of Y such that
A11: x in EqClass(y,CompF(Q,G)) and
A12: B_INF(a,CompF(P,G)).x <> TRUE;
    consider z being Element of Y such that
A13: z in EqClass(x,CompF(P,G)) and
A14: a.z <> TRUE by A12,BVFUNC_1:def 16;
A15: [x,y] in ERl '/\' B by A3,A11,Th5;
    [z,x] in ERl '/\' A by A1,A13,Th5;
    then [z,y] in (ERl '/\' B)*ERl '/\' A by A4,A15,RELAT_1:def 8;
    then consider u being object such that
A16: [z,u] in ERl '/\' B and
A17: [u,y] in ERl '/\' A by RELAT_1:def 8;
    u in field ERl '/\' B by A16,RELAT_1:15;
    then reconsider u as Element of Y by ORDERS_1:12;
    z in EqClass(u,CompF(Q,G)) by A3,A16,Th5;
    then
A18: B_INF(a,CompF(Q,G)).u = FALSE by A14,BVFUNC_1:def 16;
    u in EqClass(y,CompF(P,G)) by A1,A17,Th5;
    hence thesis by A18,BVFUNC_1:def 16;
  end;
  thus All(All(a,P,G),Q,G) = B_INF(All(a,P,G),CompF(Q,G)) by BVFUNC_2:def 9
    .= B_INF( B_INF(a,CompF(P,G)),CompF(Q,G)) by BVFUNC_2:def 9
    .= B_INF( B_INF(a,CompF(Q,G)),CompF(P,G)) by A5,BVFUNC_1:def 16
    .= B_INF(All(a,Q,G),CompF(P,G)) by BVFUNC_2:def 9
    .= All(All(a,Q,G),P,G) by BVFUNC_2:def 9;
end;
