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
  'not' (All(a,PA,G) '&' All(b,PA,G)) = Ex('not' a,PA,G) 'or' Ex('not' b
  , PA, G )
proof
A1: All(b,PA,G) = B_INF(b,CompF(PA,G)) by BVFUNC_2:def 9;
A2: All(a,PA,G) = B_INF(a,CompF(PA,G)) by BVFUNC_2:def 9;
A3: Ex('not' a,PA,G) 'or' Ex('not' b,PA,G) '<' 'not' (All(a,PA,G) '&' All(b
  ,PA,G))
  proof
    let z be Element of Y;
A4: (Ex('not' a,PA,G) 'or' Ex('not' b,PA,G)).z =Ex('not' a,PA,G).z 'or'
    Ex('not' b,PA,G).z by BVFUNC_1:def 4;
A5: Ex('not' b,PA,G).z=TRUE or Ex('not' b,PA,G).z=FALSE by XBOOLEAN:def 3;
    assume
A6: (Ex('not' a,PA,G) 'or' Ex('not' b,PA,G)).z=TRUE;
    per cases by A6,A4,A5,BINARITH:3;
    suppose
A7:   Ex('not' a,PA,G).z=TRUE;
      now
        assume not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) &
        ('not' a).x=TRUE);
        then B_SUP('not' a,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:   ('not' a).x1=TRUE;
      'not' a.x1=TRUE by A9,MARGREL1:def 19;
      then
A10:  a.x1=FALSE by MARGREL1:11;
      thus ('not' (All(a,PA,G) '&' All(b,PA,G))).z ='not' ((All(a,PA,G) '&'
      All(b,PA,G))).z by MARGREL1:def 19
        .='not' (All(a,PA,G).z '&' All(b,PA,G).z) by MARGREL1:def 20
        .='not' (FALSE '&' All(b,PA,G).z) by A2,A8,A10,BVFUNC_1:def 16
        .='not' (FALSE) by MARGREL1:12
        .=TRUE by MARGREL1:11;
    end;
    suppose
A11:  Ex('not' b,PA,G).z=TRUE;
      now
        assume not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) &
        ('not' b).x=TRUE);
        then B_SUP('not' b,CompF(PA,G)).z = FALSE by BVFUNC_1:def 17;
        hence contradiction by A11,BVFUNC_2:def 10;
      end;
      then consider x1 being Element of Y such that
A12:  x1 in EqClass(z,CompF(PA,G)) and
A13:  ('not' b).x1=TRUE;
      'not' b.x1=TRUE by A13,MARGREL1:def 19;
      then
A14:  b.x1=FALSE by MARGREL1:11;
      thus ('not' (All(a,PA,G) '&' All(b,PA,G))).z ='not' ((All(a,PA,G) '&'
      All(b,PA,G))).z by MARGREL1:def 19
        .='not' (All(a,PA,G).z '&' All(b,PA,G).z) by MARGREL1:def 20
        .='not' (All(a,PA,G).z '&' FALSE) by A1,A12,A14,BVFUNC_1:def 16
        .='not' (FALSE) by MARGREL1:12
        .=TRUE by MARGREL1:11;
    end;
  end;
  'not' (All(a,PA,G) '&' All(b,PA,G)) '<' (Ex('not' a,PA,G) 'or' Ex('not'
  b,PA,G))
  proof
    let z be Element of Y;
    assume ('not' (All(a,PA,G) '&' All(b,PA,G))).z=TRUE;
    then
A15: 'not' ((All(a,PA,G) '&' All(b,PA,G))).z=TRUE by MARGREL1:def 19;
    (All(a,PA,G) '&' All(b,PA,G)).z =All(a,PA,G).z '&' All(b,PA,G).z by
MARGREL1:def 20;
    then
A16: All(a,PA,G).z '&' All(b,PA,G).z=FALSE by A15,MARGREL1:11;
    per cases by A16,MARGREL1:12;
    suppose
      All(a,PA,G).z=FALSE;
      then consider x1 being Element of Y such that
A17:  x1 in EqClass(z,CompF(PA,G)) and
A18:  a.x1<>TRUE by A2,BVFUNC_1:def 16;
      a.x1=FALSE by A18,XBOOLEAN:def 3;
      then 'not' a.x1=TRUE by MARGREL1:11;
      then ('not' a).x1=TRUE by MARGREL1:def 19;
      then B_SUP('not' a,CompF(PA,G)).z = TRUE by A17,BVFUNC_1:def 17;
      then Ex('not' a,PA,G).z =TRUE by BVFUNC_2:def 10;
      hence
      (Ex('not' a,PA,G) 'or' Ex('not' b,PA,G)).z =TRUE 'or' Ex('not' b,PA
      ,G).z by BVFUNC_1:def 4
        .=TRUE by BINARITH:10;
    end;
    suppose
      All(b,PA,G).z=FALSE;
      then consider x1 being Element of Y such that
A19:  x1 in EqClass(z,CompF(PA,G)) and
A20:  b.x1<>TRUE by A1,BVFUNC_1:def 16;
      b.x1=FALSE by A20,XBOOLEAN:def 3;
      then 'not' b.x1=TRUE by MARGREL1:11;
      then ('not' b).x1=TRUE by MARGREL1:def 19;
      then B_SUP('not' b,CompF(PA,G)).z = TRUE by A19,BVFUNC_1:def 17;
      then Ex('not' b,PA,G).z =TRUE by BVFUNC_2:def 10;
      hence
      (Ex('not' a,PA,G) 'or' Ex('not' b,PA,G)).z =Ex('not' a,PA,G).z 'or'
      TRUE by BVFUNC_1:def 4
        .=TRUE by BINARITH:10;
    end;
  end;
  hence thesis by A3,BVFUNC_1:15;
end;
