reserve Y for non empty set;

theorem
  for u being Function of Y,BOOLEAN, G being Subset of PARTITIONS(
Y), PA being a_partition of Y st u is_independent_of PA,G holds Ex(u,PA,G) '<'
  u
proof
  let u be Function of Y,BOOLEAN;
  let G be Subset of PARTITIONS(Y);
  let PA 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' u).z = TRUE
  proof
    let z be Element of Y;
A2: (Ex(u,PA,G) 'imp' u).z ='not' Ex(u,PA,G).z 'or' u.z by BVFUNC_1:def 8;
A3: 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
        u.z=TRUE;
        hence thesis by A2;
      end;
      case
        u.z=FALSE;
        then
        not ex x1 being Element of Y st x1 in EqClass(z,CompF(PA,G)) & u.
        x1=TRUE by A1,A3;
        then B_SUP(u,CompF(PA,G)).z = FALSE by BVFUNC_1:def 17;
        hence thesis by A2;
      end;
    end;
    hence thesis;
  end;
  then Ex(u,PA,G) 'imp' u = I_el(Y) by BVFUNC_1:def 11;
  hence thesis by BVFUNC_1:16;
end;
