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
  for a being Function of Y,BOOLEAN, G being Subset of PARTITIONS(
Y), P,Q being a_partition of Y st G is independent holds Ex(All(a,P,G),Q,G) '<'
  All(Ex(a,Q,G),P,G)
proof
  let a be Function of Y,BOOLEAN, G be Subset of PARTITIONS(Y), P,Q be
  a_partition of Y such that
A1: G is independent;
  set A = G \ {P}, B = G \ {Q};
  A c= G & B c= G by XBOOLE_1:36;
  then
A2: ERl('/\'A)*ERl '/\'B = ERl ('/\'B)*ERl '/\'A by A1,Th14;
A3: CompF(P,G) = '/\' A by BVFUNC_2:def 7;
A4: Ex(All(a,P,G),Q,G) = B_SUP(All(a,P,G),CompF(Q,G)) by BVFUNC_2:def 10;
A5: CompF(Q,G) = '/\' B by BVFUNC_2:def 7;
  let x being Element of Y such that
A6: Ex(All(a,P,G),Q,G).x = TRUE;
A7: for z being Element of Y st z in EqClass(x,CompF(P,G)) holds Ex(a,Q,G).
  z=TRUE
  proof
    let z be Element of Y;
    consider y being Element of Y such that
A8: y in EqClass(x,CompF(Q,G)) and
A9: All(a,P,G).y=TRUE by A6,A4,BVFUNC_1:def 17;
    assume z in EqClass(x,CompF(P,G));
    then [z,x] in ERl '/\' A by A3,Th5;
    then
A10: [x,z] in ERl '/\' A by EQREL_1:6;
    [y,x] in ERl '/\' B by A5,A8,Th5;
    then [y,z] in (ERl '/\' A)*ERl '/\' B by A2,A10,RELAT_1:def 8;
    then consider u being object such that
A11: [y,u] in ERl '/\' A and
A12: [u,z] in ERl '/\' B by RELAT_1:def 8;
    u in field ERl '/\' B by A12,RELAT_1:15;
    then reconsider u as Element of Y by ORDERS_1:12;
    [u,y] in ERl '/\' A by A11,EQREL_1:6;
    then All(a,P,G) = B_INF(a,CompF(P,G)) & u in EqClass(y,CompF(P,G)) by A3
,Th5,BVFUNC_2:def 9;
    then
A13: a.u=TRUE by A9,BVFUNC_1:def 16;
    Ex(a,Q,G) = B_SUP(a,CompF(Q,G)) & u in EqClass(z,CompF(Q,G)) by A5,A12,Th5,
BVFUNC_2:def 10;
    hence thesis by A13,BVFUNC_1:def 17;
  end;
  All(Ex(a,Q,G),P,G) = B_INF(Ex(a,Q,G),CompF(P,G)) by BVFUNC_2:def 9;
  hence thesis by A7,BVFUNC_1:def 16;
end;
