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

theorem Th30:
  for PA being a_partition of Y st u is_independent_of PA,G holds
  All(u 'eqv' a,PA,G) '<' u 'eqv' All(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;
  assume
A3: All(u 'eqv' a,PA,G).z= TRUE;
A4: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  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: All(a,PA,G).z=TRUE & u.z=TRUE by BVFUNC_1:def 16,EQREL_1:def 6;
    (u 'eqv' All(a,PA,G)).z ='not'(u.z 'xor' All(a,PA,G).z) by BVFUNC_1:def 9
      .=TRUE by A1,A5,MARGREL1:13;
    hence thesis;
  end;
  suppose
A6: (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
A7: x1 in EqClass(z,CompF(PA,G)) and
A8: a.x1<>TRUE;
A9: u.x1=TRUE by A6,A7;
A10: a.x1=FALSE by A8,XBOOLEAN:def 3;
    (u 'eqv' a).x1 ='not'(u.x1 'xor' a.x1) by BVFUNC_1:def 9
      .='not'(FALSE 'or' TRUE) by A1,A9,A10
      .=FALSE by BINARITH:3,MARGREL1:11;
    hence thesis by A3,A7,BVFUNC_1:def 16;
  end;
  suppose
A11: not (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 consider x1 being Element of Y such that
A12: x1 in EqClass(z,CompF(PA,G)) and
A13: u.x1<>TRUE;
A14: a.x1=TRUE by A11,A12;
A15: u.x1=FALSE by A13,XBOOLEAN:def 3;
    (u 'eqv' a).x1 ='not'(u.x1 'xor' a.x1) by BVFUNC_1:def 9
      .='not'(TRUE 'or' FALSE) by A1,A15,A14
      .=FALSE by BINARITH:3,MARGREL1:11;
    hence thesis by A3,A12,BVFUNC_1:def 16;
  end;
  suppose
A16: not (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
A17: x1 in EqClass(z,CompF(PA,G)) and
A18: u.x1<>TRUE;
    u.x1=u.z by A2,A4,A17,BVFUNC_1:def 15;
    then
A19: u.z=FALSE by A18,XBOOLEAN:def 3;
A20: All(a,PA,G).z = FALSE by A16,BVFUNC_1:def 16;
    (u 'eqv' All(a,PA,G)).z ='not'(u.z 'xor' All(a,PA,G).z) by BVFUNC_1:def 9
      .=TRUE by A20,A19,MARGREL1:11,13;
    hence thesis;
  end;
end;
