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
  Ex(a,PA,G) 'imp' All(b,PA,G) '<' All(a 'imp' b,PA,G)
proof
  let z be Element of Y;
  assume (Ex(a,PA,G) 'imp' All(b,PA,G)).z=TRUE;
  then
A1: ('not' Ex(a,PA,G).z) 'or' All(b,PA,G).z=TRUE by BVFUNC_1:def 8;
  per cases;
  suppose
A2: for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds b.x= TRUE;
    for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds (a 'imp'
    b).x=TRUE
    proof
      let x be Element of Y;
      assume
A3:   x in EqClass(z,CompF(PA,G));
      thus (a 'imp' b).x=('not' a.x) 'or' b.x by BVFUNC_1:def 8
        .=('not' a.x) 'or' TRUE by A2,A3
        .=TRUE by BINARITH:10;
    end;
    then B_INF(a 'imp' b,CompF(PA,G)).z = TRUE by BVFUNC_1:def 16;
    hence thesis by BVFUNC_2:def 9;
  end;
  suppose
A4: (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a.x=TRUE
) & not (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds b.x=TRUE
    );
    then B_SUP(a,CompF(PA,G)).z = TRUE by BVFUNC_1:def 17;
    then Ex(a,PA,G).z=TRUE by BVFUNC_2:def 10;
    then
A5: 'not' Ex(a,PA,G).z=FALSE by MARGREL1:11;
    B_INF(b,CompF(PA,G)).z = FALSE by A4,BVFUNC_1:def 16;
    then All(b,PA,G).z=FALSE by BVFUNC_2:def 9;
    hence thesis by A1,A5;
  end;
  suppose
A6: not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a.x
=TRUE) & not (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds b.x
    =TRUE);
    now
      let x be Element of Y;
      assume x in EqClass(z,CompF(PA,G));
      then
A7:   a.x<>TRUE by A6;
      thus (a 'imp' b).x =('not' a.x) 'or' b.x by BVFUNC_1:def 8
        .=('not' FALSE) 'or' b.x by A7,XBOOLEAN:def 3
        .=TRUE 'or' b.x by MARGREL1:11
        .=TRUE by BINARITH:10;
    end;
    then B_INF(a 'imp' b,CompF(PA,G)).z = TRUE by BVFUNC_1:def 16;
    hence thesis by BVFUNC_2:def 9;
  end;
end;
