reserve Y for non empty set,
  G for Subset of PARTITIONS(Y),
  a,b,c,u for Function of Y,BOOLEAN,
  PA for a_partition of Y;

theorem
  u is_independent_of PA,G implies Ex(a 'imp' u,PA,G) '<' (All(a,PA,G) 'imp' u)
proof
  assume u is_independent_of PA,G;
  then
A1: u is_dependent_of CompF(PA,G) by BVFUNC_2:def 8;
  let z be Element of Y;
A2: z in EqClass(z,CompF(PA,G)) & EqClass(z,CompF(PA,G)) in CompF(PA,G) by
EQREL_1:def 6;
  assume
A3: Ex(a 'imp' u,PA,G).z=TRUE;
  now
    assume not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & (a
    'imp' u).x=TRUE);
    then B_SUP(a 'imp' u,CompF(PA,G)).z = FALSE by BVFUNC_1:def 17;
    hence contradiction by A3,BVFUNC_2:def 10;
  end;
  then consider x1 being Element of Y such that
A4: x1 in EqClass(z,CompF(PA,G)) and
A5: (a 'imp' u).x1=TRUE;
A6: ('not' a.x1) 'or' u.x1=TRUE by A5,BVFUNC_1:def 8;
A7: ('not' a.x1)=TRUE or ('not' a.x1)=FALSE by XBOOLEAN:def 3;
  per cases by A6,A7,BINARITH:3;
  suppose
    ('not' a.x1)=TRUE;
    then a.x1=FALSE by MARGREL1:11;
    then B_INF(a,CompF(PA,G)).z = FALSE by A4,BVFUNC_1:def 16;
    then All(a,PA,G).z=FALSE by BVFUNC_2:def 9;
    hence (All(a,PA,G) 'imp' u).z =('not' FALSE) 'or' u.z by BVFUNC_1:def 8
      .=TRUE 'or' u.z by MARGREL1:11
      .=TRUE by BINARITH:10;
  end;
  suppose
A8: u.x1=TRUE;
    u.x1 = u.z by A1,A4,A2;
    hence (All(a,PA,G) 'imp' u).z =('not' All(a,PA,G).z) 'or' TRUE by A8,
BVFUNC_1:def 8
      .=TRUE by BINARITH:10;
  end;
end;
