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,PA,G) 'imp' Ex(b,PA,G) = Ex(a 'imp' b,PA,G)
proof
A1: All(a,PA,G) = B_INF(a,CompF(PA,G)) by BVFUNC_2:def 9;
A2: All(a,PA,G) 'imp' Ex(b,PA,G) '<' Ex(a 'imp' b,PA,G)
  proof
    let z be Element of Y;
A3: ('not' All(a,PA,G).z)=TRUE or ('not' All(a,PA,G).z)=FALSE by XBOOLEAN:def 3
;
    assume (All(a,PA,G) 'imp' Ex(b,PA,G)).z=TRUE;
    then
A4: ('not' All(a,PA,G).z) 'or' Ex(b,PA,G).z=TRUE by BVFUNC_1:def 8;
    per cases by A4,A3,BINARITH:3;
    suppose
      ('not' All(a,PA,G).z)=TRUE;
      then All(a,PA,G).z=FALSE by MARGREL1:11;
      then consider x1 being Element of Y such that
A5:   x1 in EqClass(z,CompF(PA,G)) and
A6:   a.x1<>TRUE by A1,BVFUNC_1:def 16;
      (a 'imp' b).x1 =('not' a.x1) 'or' b.x1 by BVFUNC_1:def 8
        .=('not' FALSE) 'or' b.x1 by A6,XBOOLEAN:def 3
        .=TRUE 'or' b.x1 by MARGREL1:11
        .=TRUE by BINARITH:10;
      then B_SUP(a 'imp' b,CompF(PA,G)).z = TRUE by A5,BVFUNC_1:def 17;
      hence thesis by BVFUNC_2:def 10;
    end;
    suppose
A7:   Ex(b,PA,G).z=TRUE;
      now
        assume not (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 = FALSE by BVFUNC_1:def 17;
        hence contradiction by A7,BVFUNC_2:def 10;
      end;
      then consider x1 being Element of Y such that
A8:   x1 in EqClass(z,CompF(PA,G)) and
A9:   b.x1=TRUE;
      (a 'imp' b).x1 =('not' a.x1) 'or' TRUE by A9,BVFUNC_1:def 8
        .=TRUE by BINARITH:10;
      then B_SUP(a 'imp' b,CompF(PA,G)).z = TRUE by A8,BVFUNC_1:def 17;
      hence thesis by BVFUNC_2:def 10;
    end;
  end;
  Ex(a 'imp' b,PA,G) '<' All(a,PA,G) 'imp' Ex(b,PA,G)
  proof
    let z be Element of Y;
    assume
A10: Ex(a 'imp' b,PA,G).z=TRUE;
    now
      assume not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & (
      a 'imp' b).x=TRUE);
      then B_SUP(a 'imp' b,CompF(PA,G)).z = FALSE by BVFUNC_1:def 17;
      hence contradiction by A10,BVFUNC_2:def 10;
    end;
    then consider x1 being Element of Y such that
A11: x1 in EqClass(z,CompF(PA,G)) and
A12: (a 'imp' b).x1=TRUE;
A13: ('not' a.x1) 'or' b.x1=TRUE by A12,BVFUNC_1:def 8;
A14: ('not' a.x1)=TRUE or ('not' a.x1)=FALSE by XBOOLEAN:def 3;
    per cases by A13,A14,BINARITH:3;
    suppose
      ('not' a.x1)=TRUE;
      then a.x1=FALSE by MARGREL1:11;
      then B_INF(a,CompF(PA,G)).z = FALSE by A11,BVFUNC_1:def 16;
      hence (All(a,PA,G) 'imp' Ex(b,PA,G)).z =('not' FALSE) 'or' Ex(b,PA,G).z
      by A1,BVFUNC_1:def 8
        .=TRUE 'or' Ex(b,PA,G).z by MARGREL1:11
        .=TRUE by BINARITH:10;
    end;
    suppose
      b.x1=TRUE;
      then B_SUP(b,CompF(PA,G)).z = TRUE by A11,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;
  end;
  hence thesis by A2,BVFUNC_1:15;
end;
