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
  (a 'imp' b) '<' (All(a,PA,G) 'imp' Ex(b,PA,G))
proof
  let z be Element of Y;
A1: ('not' a.z)=TRUE or ('not' a.z)=FALSE by XBOOLEAN:def 3;
A2: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  assume (a 'imp' b).z=TRUE;
  then
A3: ('not' a.z) 'or' b.z=TRUE by BVFUNC_1:def 8;
  per cases by A3,A1,BINARITH:3;
  suppose
    'not' a.z=TRUE;
    then a.z=FALSE by MARGREL1:11;
    then B_INF(a,CompF(PA,G)).z = FALSE by A2,BVFUNC_1:def 16;
    then All(a,PA,G).z=FALSE by BVFUNC_2:def 9;
    hence (All(a,PA,G) 'imp' Ex(b,PA,G)).z =('not' FALSE) 'or' Ex(b,PA,G).z by
BVFUNC_1:def 8
      .=TRUE 'or' Ex(b,PA,G).z by MARGREL1:11
      .=TRUE by BINARITH:10;
  end;
  suppose
    b.z=TRUE;
    then B_SUP(b,CompF(PA,G)).z = TRUE by A2,BVFUNC_1:def 17;
    then Ex(b,PA,G).z=TRUE by BVFUNC_2:def 10;
    hence (All(a,PA,G) 'imp' Ex(b,PA,G)).z =('not' All(a,PA,G).z) 'or' TRUE by
BVFUNC_1:def 8
      .=TRUE by BINARITH:10;
  end;
end;
