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

theorem Th22:
  for PA being a_partition of Y st u is_independent_of PA,G holds
  All(u 'or' a,PA,G) = u 'or' All(a,PA,G)
proof
  let PA be a_partition of Y;
  assume
A1: u is_independent_of PA,G;
    let z be Element of Y;
A2: (u 'or' B_INF(a,CompF(PA,G))).z = u.z 'or' B_INF(a,CompF(PA,G)).z by
BVFUNC_1:def 4;
    per cases;
    suppose
A3:   for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a. x=TRUE;
A4:   for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds (u
      'or' a).x=TRUE
      proof
        let x be Element of Y;
        assume
A5:     x in EqClass(z,CompF(PA,G));
        (u 'or' a).x = u.x 'or' a.x by BVFUNC_1:def 4
          .= u.x 'or' TRUE by A3,A5
          .= TRUE by BINARITH:10;
        hence thesis;
      end;
      B_INF(a,CompF(PA,G)).z = TRUE by A3,BVFUNC_1:def 16;
      then (u 'or' B_INF(a,CompF(PA,G))).z = TRUE by A2,BINARITH:10;
      hence thesis by A4,BVFUNC_1:def 16;
    end;
    suppose
A6:  not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
holds a.x=TRUE) & for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds
      u.x=TRUE;
A7:  for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds (u
      'or' a).x=TRUE
      proof
        let x be Element of Y;
        assume
A8:    x in EqClass(z,CompF(PA,G));
        (u 'or' a).x = u.x 'or' a.x by BVFUNC_1:def 4
          .= TRUE 'or' a.x by A6,A8
          .= TRUE by BINARITH:10;
        hence thesis;
      end;
      (u 'or' B_INF(a,CompF(PA,G))).z = TRUE 'or' B_INF(a,CompF(PA,G)).z
      by A2,A6,EQREL_1:def 6;
      then (u 'or' B_INF(a,CompF(PA,G))).z = TRUE by BINARITH:10;
      hence thesis by A7,BVFUNC_1:def 16;
    end;
    suppose
A9:  not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
holds a.x=TRUE) & not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
      holds u.x=TRUE);
      then consider x2 being Element of Y such that
A10:  x2 in EqClass(z,CompF(PA,G)) and
A11:  u.x2<>TRUE;
      consider x1 being Element of Y such that
A12:  x1 in EqClass(z,CompF(PA,G)) and
A13:  a.x1<>TRUE by A9;
      u.x1 = u.x2 by A1,A12,A10,BVFUNC_1:def 15;
      then
A14:  u.x1 = FALSE by A11,XBOOLEAN:def 3;
A15:  B_INF(a,CompF(PA,G)).z = FALSE by A9,BVFUNC_1:def 16;
      z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
      then
A16:  u.x1 = u.z by A1,A12,BVFUNC_1:def 15;
      a.x1 = FALSE by A13,XBOOLEAN:def 3;
      then (u 'or' a).x1 = FALSE 'or' FALSE by A14,BVFUNC_1:def 4;
      hence thesis by A2,A15,A12,A14,A16,BVFUNC_1:def 16;
    end;
end;
