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

theorem Th38:
  for PA being a_partition of Y st u is_independent_of PA,G holds
  u 'xor' Ex(a,PA,G) '<' Ex(u 'xor' a,PA,G)
proof
  let PA be a_partition of Y;
A1: 'not' FALSE=TRUE by MARGREL1:11;
  assume
A2: u is_independent_of PA,G;
  let z be Element of Y;
A3: (u 'xor' Ex(a,PA,G)).z =u.z 'xor' Ex(a,PA,G).z by BVFUNC_1:def 5
    .=('not' u.z '&' Ex(a,PA,G).z) 'or' (u.z '&' 'not' Ex(a,PA,G).z);
A4: (u.z '&' 'not' Ex(a,PA,G).z)=TRUE or (u.z '&' 'not' Ex(a,PA,G).z)=FALSE
  by XBOOLEAN:def 3;
A5: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  assume
A6: (u 'xor' Ex(a,PA,G)).z= TRUE;
  now
    per cases by A6,A3,A4,BINARITH:3;
    case
A7:   'not' u.z '&' Ex(a,PA,G).z=TRUE;
      then Ex(a,PA,G).z=TRUE by MARGREL1:12;
      then consider x1 being Element of Y such that
A8:   x1 in EqClass(z,CompF(PA,G)) and
A9:   a.x1=TRUE by BVFUNC_1:def 17;
A10:  u.z=u.x1 by A2,A5,A8,BVFUNC_1:def 15;
A11:  'not' u.z=TRUE by A7,MARGREL1:12;
      (u 'xor' a).x1 =u.x1 'xor' a.x1 by BVFUNC_1:def 5
        .= TRUE 'or' FALSE by A11,A9,A10,MARGREL1:11
        .= TRUE by BINARITH:10;
      hence thesis by A8,BVFUNC_1:def 17;
    end;
    case
A12:  u.z '&' 'not' Ex(a,PA,G).z=TRUE;
      then 'not' Ex(a,PA,G).z=TRUE by MARGREL1:12;
      then a.z<>TRUE by A5,BVFUNC_1:def 17,MARGREL1:11;
      then
A13:  a.z=FALSE by XBOOLEAN:def 3;
A14:  u.z=TRUE by A12,MARGREL1:12;
      (u 'xor' a).z = u.z 'xor' a.z by BVFUNC_1:def 5
        .= FALSE 'or' TRUE by A1,A14,A13
        .= TRUE by BINARITH:10;
      hence thesis by A5,BVFUNC_1:def 17;
    end;
  end;
  hence thesis;
end;
