reserve Y for non empty set;

theorem
  for u being Function of Y,BOOLEAN, G being Subset of PARTITIONS(
  Y), PA,PB being a_partition of Y st u is_independent_of PA,G holds Ex(u,PA,G)
  '<' Ex(u,PB,G)
proof
  let u be Function of Y,BOOLEAN;
  let G be Subset of PARTITIONS(Y);
  let PA,PB be a_partition of Y;
  assume u is_independent_of PA,G;
  then
A1: u is_dependent_of CompF(PA,G);
  for z being Element of Y holds (Ex(u,PA,G) 'imp' Ex(u,PB,G)).z = TRUE
  proof
    let z be Element of Y;
A2: z in EqClass(z,CompF(PB,G)) by EQREL_1:def 6;
A3: (Ex(u,PA,G) 'imp' Ex(u,PB,G)).z ='not' Ex(u,PA,G).z 'or' Ex(u,PB,G).z
    by BVFUNC_1:def 8;
A4: z in EqClass(z,CompF(PA,G)) & EqClass(z,CompF(PA,G)) in CompF(PA,G) by
EQREL_1:def 6;
    now
      per cases by XBOOLEAN:def 3;
      case
        Ex(u,PB,G).z=TRUE;
        hence thesis by A3;
      end;
      case
        Ex(u,PB,G).z=FALSE;
        then u.z<>TRUE by A2,BVFUNC_1:def 17;
        then
        not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & u.x
        =TRUE) by A1,A4;
        then (B_SUP(u,CompF(PA,G))).z = FALSE by BVFUNC_1:def 17;
        hence thesis by A3;
      end;
    end;
    hence thesis;
  end;
  then Ex(u,PA,G) 'imp' Ex(u,PB,G) = I_el(Y) by BVFUNC_1:def 11;
  hence thesis by BVFUNC_1:16;
end;
