reserve Y for non empty set,
  G for Subset of PARTITIONS(Y);

theorem
  for a being Function of Y,BOOLEAN, PA being a_partition of Y
  holds All(a,PA,G) is_dependent_of CompF(PA,G)
proof
  let a be Function of Y,BOOLEAN;
  let PA be a_partition of Y;
  let F be set;
  assume
A1: F in CompF(PA,G);
  thus for x1,x2 being set st x1 in F & x2 in F holds All(a,PA,G).x1=All(a,PA,
  G).x2
  proof
    let x1,x2 be set;
    assume
A2: x1 in F & x2 in F;
    then reconsider x1,x2 as Element of Y by A1;
A3: x2 in EqClass(x2,CompF(PA,G)) by EQREL_1:def 6;
    F = EqClass(x2,CompF(PA,G)) or F misses EqClass(x2,CompF(PA,G)) by A1,
EQREL_1:def 4;
    then
A4: EqClass(x1,CompF(PA,G)) = EqClass(x2,CompF(PA,G)) by A2,A3,EQREL_1:def 6
,XBOOLE_0:3;
    per cases;
    suppose
A5:   (for x being Element of Y st x in EqClass(x1,CompF(PA,G)) holds
a.x=TRUE) & for x being Element of Y st x in EqClass(x2,CompF(PA,G)) holds a.x=
      TRUE;
      then All(a,PA,G).x2=TRUE by BVFUNC_1:def 16;
      hence thesis by A5,BVFUNC_1:def 16;
    end;
    suppose
      (for x being Element of Y st x in EqClass(x1,CompF(PA,G)) holds
a.x=TRUE) & not (for x being Element of Y st x in EqClass(x2,CompF(PA,G)) holds
      a.x=TRUE);
      hence thesis by A4;
    end;
    suppose
      not (for x being Element of Y st x in EqClass(x1,CompF(PA,G))
holds a.x=TRUE) & for x being Element of Y st x in EqClass(x2,CompF(PA,G))
      holds a.x=TRUE;
      hence thesis by A4;
    end;
    suppose
A6:   not (for x being Element of Y st x in EqClass(x1,CompF(PA,G))
holds a.x=TRUE) & not (for x being Element of Y st x in EqClass(x2,CompF(PA,G))
      holds a.x=TRUE);
      then All(a,PA,G).x2=FALSE by BVFUNC_1:def 16;
      hence thesis by A6,BVFUNC_1:def 16;
    end;
  end;
end;
