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

theorem Th28:
  for PA being a_partition of Y st u is_independent_of PA,G holds
  All(u 'xor' a,PA,G) '<' u 'xor' 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;
  assume
A2: All(u 'xor' a,PA,G).z= TRUE;
A3: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
A4: 'not' FALSE=TRUE & (u 'xor' All(a,PA,G)).z = All(a,PA,G).z 'xor' u.z by
BVFUNC_1:def 5,MARGREL1:11;
  per cases;
  suppose
    (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds u.x
=TRUE) & for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a.x=TRUE
    ;
    then
A5: u.z=TRUE & a.z=TRUE by EQREL_1:def 6;
A6: FALSE '&' TRUE = FALSE by MARGREL1:12;
    (u 'xor' a).z =a.z 'xor' u.z by BVFUNC_1:def 5
      .=FALSE by A5,A6,MARGREL1:11;
    hence thesis by A2,A3,BVFUNC_1:def 16;
  end;
  suppose
A7: (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds u.
x=TRUE) & not (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a.
    x=TRUE);
    then consider x1 being Element of Y such that
A8: x1 in EqClass(z,CompF(PA,G)) and
    a.x1<>TRUE;
A9: u.x1=TRUE by A7,A8;
A10: All(a,PA,G).z = FALSE by A7,BVFUNC_1:def 16;
    u.z=u.x1 by A1,A3,A8,BVFUNC_1:def 15;
    then (u 'xor' All(a,PA,G)).z =TRUE 'or' FALSE by A4,A10,A9
      .=TRUE by BINARITH:3;
    hence thesis;
  end;
  suppose
    not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
    holds u.x=TRUE);
    then consider x1 being Element of Y such that
A11: x1 in EqClass(z,CompF(PA,G)) and
A12: u.x1<>TRUE;
    now
      per cases;
      suppose
A13:    for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a.x=TRUE;
        u.z=u.x1 by A1,A3,A11,BVFUNC_1:def 15;
        then
A14:    u.z=FALSE by A12,XBOOLEAN:def 3;
        All(a,PA,G).z = TRUE by A13,BVFUNC_1:def 16;
        then (u 'xor' All(a,PA,G)).z =FALSE 'or' TRUE by A4,A14
          .=TRUE by BINARITH:3;
        hence thesis;
      end;
      suppose
        not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
        holds a.x=TRUE);
        then consider x2 being Element of Y such that
A15:    x2 in EqClass(z,CompF(PA,G)) and
A16:    a.x2<>TRUE;
A17:    a.x2=FALSE by A16,XBOOLEAN:def 3;
        u.x1=u.x2 by A1,A11,A15,BVFUNC_1:def 15;
        then
A18:    u.x2=FALSE by A12,XBOOLEAN:def 3;
        (u 'xor' a).x2 =a.x2 'xor' u.x2 by BVFUNC_1:def 5
          .=FALSE by A18,A17,MARGREL1:12;
        hence thesis by A2,A15,BVFUNC_1:def 16;
      end;
    end;
    hence thesis;
  end;
end;
