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
  All(a 'imp' b,PA,G) '<' All(a,PA,G) 'imp' Ex(b,PA,G)
proof
A1: All(a,PA,G) = B_INF(a,CompF(PA,G)) by BVFUNC_2:def 9;
  let z be Element of Y;
  assume
A2: All(a 'imp' b,PA,G).z=TRUE;
A3: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  per cases;
  suppose
    ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & b.x=TRUE;
    then B_SUP(b,CompF(PA,G)).z = TRUE by 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;
  suppose
    (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a.x
=TRUE) & not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & b.x=TRUE
    );
    then
A4: b.z<>TRUE & a.z=TRUE by A3;
    (a 'imp' b).z =('not' a.z) 'or' b.z by BVFUNC_1:def 8
      .=('not' TRUE) 'or' FALSE by A4,XBOOLEAN:def 3
      .=FALSE 'or' FALSE by MARGREL1:11
      .=FALSE;
    hence thesis by A2,A3,Lm2;
  end;
  suppose
A5: not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
holds a.x=TRUE) & not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) &
    b.x=TRUE);
    thus (All(a,PA,G) 'imp' Ex(b,PA,G)).z =('not' All(a,PA,G).z) 'or' Ex(b,PA,
    G).z by BVFUNC_1:def 8
      .=('not' FALSE) 'or' Ex(b,PA,G).z by A1,A5,BVFUNC_1:def 16
      .=TRUE 'or' Ex(b,PA,G).z by MARGREL1:11
      .=TRUE by BINARITH:10;
  end;
end;
