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

theorem Th25:
  for PA being a_partition of Y st u is_independent_of PA,G holds
  All(u '&' a,PA,G) = u '&' All(a,PA,G)
proof
  let PA be a_partition of Y;
A1: All(u '&' a,PA,G) '<' (u '&' All(a,PA,G))
  proof
    let z be Element of Y;
    assume
A2: All(u '&' a,PA,G).z= TRUE;
A3: for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds u.x=TRUE
    proof
      let x be Element of Y;
      assume x in EqClass(z,CompF(PA,G));
      then
A4:   (u '&' a).x=TRUE by A2,BVFUNC_1:def 16;
      (u '&' a).x=u.x '&' a.x by MARGREL1:def 20;
      hence thesis by A4,MARGREL1:12;
    end;
    for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a.x=TRUE
    proof
      let x be Element of Y;
      assume x in EqClass(z,CompF(PA,G));
      then
A5:   (u '&' a).x=TRUE by A2,BVFUNC_1:def 16;
      (u '&' a).x=u.x '&' a.x by MARGREL1:def 20;
      hence thesis by A5,MARGREL1:12;
    end;
    then
A6: (u '&' All(a,PA,G)).z= u.z '&' All(a,PA,G).z & All(a,PA,G).z=TRUE by
BVFUNC_1:def 16,MARGREL1:def 20;
    u.z=TRUE by A3,EQREL_1:def 6;
    hence thesis by A6;
  end;
  assume
A7: u is_independent_of PA,G;
  (u '&' All(a,PA,G)) '<' All(u '&' a,PA,G)
  proof
    let z be Element of Y;
A8: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
    assume (u '&' All(a,PA,G)).z= TRUE;
    then
A9: u.z '&' All(a,PA,G).z= TRUE by MARGREL1:def 20;
    then
A10: All(a,PA,G).z=TRUE by MARGREL1:12;
A11: u.z=TRUE by A9,MARGREL1:12;
    for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds (u '&'
    a).x=TRUE
    proof
      let x be Element of Y;
      assume x in EqClass(z,CompF(PA,G));
      then a.x=TRUE & u.x=u.z by A7,A10,A8,BVFUNC_1:def 15,def 16;
      then (u '&' a).x =TRUE '&' TRUE by A11,MARGREL1:def 20
        .=TRUE;
      hence thesis;
    end;
    hence thesis by BVFUNC_1:def 16;
  end;
  hence thesis by A1,BVFUNC_1:15;
end;
