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) '<' 'not' ('not' Ex(a '&' b,PA,G) '&' 'not' Ex(a '&' 'not'
  b,PA,G))
proof
  let z be Element of Y;
A1: ('not' Ex(a '&' 'not' b,PA,G)).z='not' (Ex(a '&' 'not' b,PA,G)).z by
MARGREL1:def 19;
A2: ('not' ('not' Ex(a '&' b,PA,G) '&' 'not' Ex(a '&' 'not' b,PA,G))).z =
'not' (('not' Ex(a '&' b,PA,G) '&' 'not' Ex(a '&' 'not' b,PA,G))).z by
MARGREL1:def 19
    .='not' (('not' Ex(a '&' b,PA,G)).z '&' ('not' Ex(a '&' 'not' b,PA,G)).z
  ) by MARGREL1:def 20
    .='not' ('not' Ex(a '&' b,PA,G).z '&' 'not' (Ex(a '&' 'not' b,PA,G)).z)
  by A1,MARGREL1:def 19;
  assume
A3: Ex(a,PA,G).z=TRUE;
  now
    assume
    not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a.x= TRUE);
    then B_SUP(a,CompF(PA,G)).z = FALSE by BVFUNC_1:def 17;
    hence contradiction by A3,BVFUNC_2:def 10;
  end;
  then consider x1 being Element of Y such that
A4: x1 in EqClass(z,CompF(PA,G)) and
A5: a.x1=TRUE;
A6: (a '&' b).x1 =TRUE '&' b.x1 by A5,MARGREL1:def 20
    .=b.x1 by MARGREL1:14;
A7: (a '&' 'not' b).x1 =TRUE '&' ('not' b).x1 by A5,MARGREL1:def 20
    .=('not' b).x1 by MARGREL1:14;
  per cases by XBOOLEAN:def 3;
  suppose
    b.x1=TRUE;
    then B_SUP(a '&' b,CompF(PA,G)).z = TRUE by A4,A6,BVFUNC_1:def 17;
    hence
    ('not' ('not' Ex(a '&' b,PA,G) '&' 'not' Ex(a '&' 'not' b,PA,G))).z =
    'not' ('not' TRUE '&' 'not' (Ex(a '&' 'not' b,PA,G)).z) by A2,
BVFUNC_2:def 10
      .='not' (FALSE '&' 'not' (Ex(a '&' 'not' b,PA,G)).z) by MARGREL1:11
      .='not' (FALSE) by MARGREL1:12
      .=TRUE by MARGREL1:11;
  end;
  suppose
    b.x1=FALSE;
    then (a '&' 'not' b).x1='not' FALSE by A7,MARGREL1:def 19;
    then (a '&' 'not' b).x1=TRUE by MARGREL1:11;
    then (B_SUP(a '&' 'not' b,CompF(PA,G))).z = TRUE by A4,BVFUNC_1:def 17;
    hence
    ('not' ('not' Ex(a '&' b,PA,G) '&' 'not' Ex(a '&' 'not' b,PA,G))).z =
    'not' ('not' Ex(a '&' b,PA,G).z '&' 'not' TRUE) by A2,BVFUNC_2:def 10
      .='not' ('not' Ex(a '&' b,PA,G).z '&' FALSE) by MARGREL1:11
      .='not' (FALSE) by MARGREL1:12
      .=TRUE by MARGREL1:11;
  end;
end;
