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(u 'imp' a,PA,G) '<' (u 'imp' Ex(a, PA,G))
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(u 'imp' a,PA,G).z=TRUE;
  now
    assume not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & (u
    'imp' a).x=TRUE);
    then B_SUP(u 'imp' a,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: (u 'imp' a).x1=TRUE;
A6: ('not' u.x1) 'or' a.x1=TRUE by A5,BVFUNC_1:def 8;
A7: ('not' u.x1)=TRUE or ('not' u.x1)=FALSE by XBOOLEAN:def 3;
  per cases by A6,A7,BINARITH:3;
  suppose
A8: ('not' u.x1)=TRUE;
    u.x1 = u.z by A1,A4,A2;
    then u.z=FALSE by A8,MARGREL1:11;
    hence (u 'imp' Ex(a,PA,G)).z =('not' FALSE) 'or' Ex(a,PA,G).z by
BVFUNC_1:def 8
      .=TRUE 'or' Ex(a,PA,G).z by MARGREL1:11
      .=TRUE by BINARITH:10;
  end;
  suppose
    a.x1=TRUE;
    then B_SUP(a,CompF(PA,G)).z = TRUE by A4,BVFUNC_1:def 17;
    then Ex(a,PA,G).z=TRUE by BVFUNC_2:def 10;
    hence (u 'imp' Ex(a,PA,G)).z =('not' u.z) 'or' TRUE by BVFUNC_1:def 8
      .=TRUE by BINARITH:10;
  end;
end;
