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 Ex(a,PA,G) 'imp' Ex(b,PA,G) '<' Ex(a 'imp' 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;
A1: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  assume (Ex(a,PA,G) 'imp' Ex(b,PA,G)).z=TRUE;
  then
A2: ('not' Ex(a,PA,G).z) 'or' Ex(b,PA,G).z=TRUE by BVFUNC_1:def 8;
A3: ('not' Ex(a,PA,G).z)=TRUE or ('not' Ex(a,PA,G).z)=FALSE by XBOOLEAN:def 3;
  now
    per cases by A2,A3,BINARITH:3;
    case
      ('not' Ex(a,PA,G).z)=TRUE;
      then
A4:   a.z<>TRUE by A1,BVFUNC_1:def 17,MARGREL1:11;
      (a 'imp' b).z=('not' a.z) 'or' b.z by BVFUNC_1:def 8
        .=TRUE 'or' b.z by A4,MARGREL1:11,XBOOLEAN:def 3
        .=TRUE by BINARITH:10;
      hence thesis by A1,BVFUNC_1:def 17;
    end;
    case
      Ex(b,PA,G).z=TRUE;
      then consider x1 being Element of Y such that
A5:   x1 in EqClass(z,CompF(PA,G)) and
A6:   b.x1=TRUE by BVFUNC_1:def 17;
      (a 'imp' b).x1 =('not' a.x1) 'or' b.x1 by BVFUNC_1:def 8
        .=TRUE by A6,BINARITH:10;
      hence thesis by A5,BVFUNC_1:def 17;
    end;
  end;
  hence thesis;
end;
