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

theorem Th34:
  for PA being a_partition of Y st u is_independent_of PA,G holds
  Ex(u '&' a,PA,G) = u '&' Ex(a,PA,G)
proof
  let PA be a_partition of Y;
  assume
A1: u is_independent_of PA,G;
A2: Ex(u '&' a,PA,G) '<' (u '&' Ex(a,PA,G))
  proof
    let z be Element of Y;
    assume Ex(u '&' a,PA,G).z= TRUE;
    then consider x1 being Element of Y such that
A3: x1 in EqClass(z,CompF(PA,G)) and
A4: (u '&' a).x1=TRUE by BVFUNC_1:def 17;
A5: u.x1 '&' a.x1=TRUE by A4,MARGREL1:def 20;
    then a.x1=TRUE by MARGREL1:12;
    then
A6: Ex(a,PA,G).z=TRUE by A3,BVFUNC_1:def 17;
    z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
    then
A7: u.z=u.x1 by A1,A3,BVFUNC_1:def 15;
    u.x1=TRUE by A5,MARGREL1:12;
    then (u '&' Ex(a,PA,G)).z =TRUE '&' TRUE by A6,A7,MARGREL1:def 20
      .=TRUE;
    hence thesis;
  end;
  (u '&' Ex(a,PA,G)) '<' Ex(u '&' a,PA,G)
  proof
    let z be Element of Y;
    assume (u '&' Ex(a,PA,G)).z= TRUE;
    then
A8: u.z '&' Ex(a,PA,G).z=TRUE by MARGREL1:def 20;
    then
A9: u.z=TRUE by MARGREL1:12;
    Ex(a,PA,G).z=TRUE by A8,MARGREL1:12;
    then consider x1 being Element of Y such that
A10: x1 in EqClass(z,CompF(PA,G)) and
A11: a.x1=TRUE by BVFUNC_1:def 17;
    z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
    then u.x1=u.z by A1,A10,BVFUNC_1:def 15;
    then (u '&' a).x1=TRUE '&' TRUE by A9,A11,MARGREL1:def 20
      .=TRUE;
    hence thesis by A10,BVFUNC_1:def 17;
  end;
  hence thesis by A2,BVFUNC_1:15;
end;
