reserve Y for non empty set,
  G for Subset of PARTITIONS(Y);

theorem
  for a,b being Function of Y,BOOLEAN, PA being a_partition of Y
  holds All(a 'imp' b,PA,G) '<' All(a,PA,G) 'imp' All(b,PA,G)
proof
  let a,b be Function of Y,BOOLEAN;
  let PA be a_partition of Y;
  let z be Element of Y;
  assume
A1: All(a 'imp' b,PA,G).z=TRUE;
A2: (All(a,PA,G) 'imp' All(b,PA,G)).z =('not' All(a,PA,G).z) 'or' All(b,PA,G
  ).z by BVFUNC_1:def 8;
  per cases;
  suppose
    (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a.x
=TRUE) & for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds b.x=TRUE
    ;
    then B_INF(b,CompF(PA,G)).z = TRUE by BVFUNC_1:def 16;
    then
    (All(a,PA,G) 'imp' All(b,PA,G)).z =('not' All(a,PA,G).z) 'or' TRUE by
BVFUNC_1:def 8
      .=TRUE by BINARITH:10;
    hence thesis;
  end;
  suppose
A3: (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a.x
=TRUE) & not (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds b.x
    =TRUE);
    then consider x1 being Element of Y such that
A4: x1 in EqClass(z,CompF(PA,G)) and
A5: b.x1<>TRUE;
A6: a.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 A1,A4,BVFUNC_1:def 16;
  end;
  suppose
    not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
holds a.x=TRUE) & for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds
    b.x=TRUE;
    then
    (All(a,PA,G) 'imp' All(b,PA,G)).z =('not' All(a,PA,G).z) 'or' TRUE by A2,
BVFUNC_1:def 16
      .=TRUE by BINARITH:10;
    hence thesis;
  end;
  suppose
    not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
holds a.x=TRUE) & not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
    holds b.x=TRUE);
    then (All(a,PA,G) 'imp' All(b,PA,G)).z
       =TRUE 'or' All(b,PA,G).z by A2,BVFUNC_1:def 16,MARGREL1:11
      .=TRUE by BINARITH:10;
    hence thesis;
  end;
end;
