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 holds All(a '&' u,PA,G) '<' Ex(a,PA,G) '&' u
proof
  let PA be a_partition of Y;
  let z be Element of Y;
A1: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  assume
A2: All(a '&' u,PA,G).z= TRUE;
A3: for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a.x=TRUE &
  u.x=TRUE
  proof
    let x be Element of Y;
    assume x in EqClass(z,CompF(PA,G));
    then (a '&' u).x=TRUE by A2,BVFUNC_1:def 16;
    then a.x '&' u.x=TRUE by MARGREL1:def 20;
    hence thesis by MARGREL1:12;
  end;
A4: (Ex(a,PA,G) '&' u).z = Ex(a,PA,G).z '&' u.z by MARGREL1:def 20;
  per cases;
  suppose
A5: for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds u.x= TRUE;
    now
      per cases;
      suppose
        ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a.x= TRUE;
        then Ex(a,PA,G).z = TRUE by BVFUNC_1:def 17;
        hence (Ex(a,PA,G) '&' u).z = TRUE '&' TRUE by A4,A5,EQREL_1:def 6
          .= TRUE;
      end;
      suppose
        not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) &
        a.x=TRUE);
        then a.z<>TRUE by EQREL_1:def 6;
        hence thesis by A3,A1;
      end;
    end;
    hence thesis;
  end;
  suppose
    not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
    holds u.x=TRUE);
    hence thesis by A3;
  end;
end;
