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

theorem
  for PA being a_partition of Y st u is_independent_of PA,G holds All(a
  'or' u,PA,G) '<' Ex(a,PA,G) 'or' u
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(a 'or' u,PA,G).z= TRUE;
A3: for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a.x=TRUE
  or u.x=TRUE
  proof
    let x be Element of Y;
    assume x in EqClass(z,CompF(PA,G));
    then (a 'or' u).x=TRUE by A2,BVFUNC_1:def 16;
    then
A4: a.x 'or' u.x=TRUE by BVFUNC_1:def 4;
    u.x= TRUE or u.x=FALSE by XBOOLEAN:def 3;
    hence thesis by A4,BINARITH:3;
  end;
A5: (Ex(a,PA,G) 'or' u).z = Ex(a,PA,G).z 'or' u.z by BVFUNC_1:def 4;
  per cases;
  suppose
    for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds u.x =TRUE;
    then (Ex(a,PA,G) 'or' u).z = Ex(a,PA,G).z 'or' TRUE by A5,EQREL_1:def 6
      .= TRUE by BINARITH:10;
    hence thesis;
  end;
  suppose
    not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
holds u.x=TRUE) & ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a.x=
    TRUE;
    then (Ex(a,PA,G) 'or' u).z = TRUE 'or' u.z by A5,BVFUNC_1:def 17
      .= TRUE by BINARITH:10;
    hence thesis;
  end;
  suppose
A6: not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
holds u.x=TRUE) & not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) &
    a.x=TRUE);
A7: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
A8: a.z<>TRUE by A6,EQREL_1:def 6;
    consider x1 being Element of Y such that
A9: x1 in EqClass(z,CompF(PA,G)) and
A10: u.x1<>TRUE by A6;
    u.x1=u.z by A1,A7,A9,BVFUNC_1:def 15;
    hence thesis by A3,A8,A10,EQREL_1:def 6;
  end;
end;
