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 Ex(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 Ex(a,PA,G).x1=Ex(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:   (ex x being Element of Y st x in EqClass(x1,CompF(PA,G)) & a.x=
TRUE) & ex x being Element of Y st x in EqClass(x2,CompF(PA,G)) & a.x=TRUE;
      then B_SUP(a,CompF(PA,G)).x1 = TRUE by BVFUNC_1:def 17;
      hence thesis by A5,BVFUNC_1:def 17;
    end;
    suppose
      (ex x being Element of Y st x in EqClass(x1,CompF(PA,G)) & a.x=
TRUE) & not (ex x being Element of Y st x in EqClass(x2,CompF(PA,G)) & a.x=TRUE
      );
      hence thesis by A4;
    end;
    suppose
      not (ex x being Element of Y st x in EqClass(x1,CompF(PA,G)) &
a.x=TRUE) & ex x being Element of Y st x in EqClass(x2,CompF(PA,G)) & a.x=TRUE;
      hence thesis by A4;
    end;
    suppose
A6:   not (ex x being Element of Y st x in EqClass(x1,CompF(PA,G)) &
a.x=TRUE) & not (ex x being Element of Y st x in EqClass(x2,CompF(PA,G)) & a.x=
      TRUE);
      then B_SUP(a,CompF(PA,G)).x1 = FALSE by BVFUNC_1:def 17;
      hence thesis by A6,BVFUNC_1:def 17;
    end;
  end;
end;
