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) '<' Ex(a,PA,G) 'imp' Ex(b,PA,G)
proof
A1: Ex(a,PA,G) = B_SUP(a,CompF(PA,G)) by BVFUNC_2:def 10;
  let z be Element of Y;
  assume
A2: All(a 'imp' b,PA,G).z=TRUE;
  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 (Ex(a,PA,G) 'imp' Ex(b,PA,G)).z =('not' Ex(a,PA,G).z) 'or' TRUE by
BVFUNC_1:def 8
      .=TRUE by BINARITH:10;
  end;
  suppose
A3: (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a.x=TRUE
) & not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & b.x=TRUE);
    then consider x1 being Element of Y such that
A4: x1 in EqClass(z,CompF(PA,G)) and
A5: a.x1=TRUE;
A6: b.x1<>TRUE by A3,A4;
    (a 'imp' b).x1 =('not' a.x1) 'or' b.x1 by BVFUNC_1:def 8
      .=('not' TRUE) 'or' FALSE by A5,A6,XBOOLEAN:def 3
      .=FALSE 'or' FALSE by MARGREL1:11
      .=FALSE;
    hence thesis by A2,A4,Lm2;
  end;
  suppose
A7: not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a.x
=TRUE) & not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & b.x=TRUE
    );
    thus (Ex(a,PA,G) 'imp' Ex(b,PA,G)).z =('not' Ex(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,A7,BVFUNC_1:def 17
      .=TRUE 'or' Ex(b,PA,G).z by MARGREL1:11
      .=TRUE by BINARITH:10;
  end;
end;
