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' (All(a 'imp' b,PA,G) '&' All(a 'imp' 'not' b,PA,G ))
proof
  let z be Element of Y;
A1: ('not' (All(a 'imp' b,PA,G) '&' All(a 'imp' 'not' b,PA,G))).z ='not' ((
  All(a 'imp' b,PA,G) '&' All(a 'imp' 'not' b,PA,G))).z by MARGREL1:def 19
    .='not' (All(a 'imp' b,PA,G).z '&' (All(a 'imp' 'not' b,PA,G)).z) by
MARGREL1:def 20;
  assume
A2: 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 A2,BVFUNC_2:def 10;
  end;
  then consider x1 being Element of Y such that
A3: x1 in EqClass(z,CompF(PA,G)) and
A4: a.x1=TRUE;
A5: (a 'imp' b).x1 ='not' TRUE 'or' b.x1 by A4,BVFUNC_1:def 8
    .=FALSE 'or' b.x1 by MARGREL1:11
    .=b.x1 by BINARITH:3;
A6: (a 'imp' 'not' b).x1 ='not' TRUE 'or' ('not' b).x1 by A4,BVFUNC_1:def 8
    .=FALSE 'or' ('not' b).x1 by MARGREL1:11
    .=('not' b).x1 by BINARITH:3;
  per cases by XBOOLEAN:def 3;
  suppose
    b.x1=TRUE;
    then (a 'imp' 'not' b).x1 ='not' TRUE by A6,MARGREL1:def 19
      .=FALSE by MARGREL1:11;
    then (B_INF(a 'imp' 'not' b,CompF(PA,G))).z = FALSE by A3,BVFUNC_1:def 16;
    hence ('not' (All(a 'imp' b,PA,G) '&' All(a 'imp' 'not' b,PA,G))).z ='not'
    (All(a 'imp' b,PA,G).z '&' FALSE) by A1,BVFUNC_2:def 9
      .='not' (FALSE) by MARGREL1:12
      .=TRUE by MARGREL1:11;
  end;
  suppose
    b.x1=FALSE;
    then B_INF(a 'imp' b,CompF(PA,G)).z = FALSE by A3,A5,BVFUNC_1:def 16;
    hence ('not' (All(a 'imp' b,PA,G) '&' All(a 'imp' 'not' b,PA,G))).z ='not'
    (FALSE '&' (All(a 'imp' 'not' b,PA,G)).z) by A1,BVFUNC_2:def 9
      .='not' (FALSE) by MARGREL1:12
      .=TRUE by MARGREL1:11;
  end;
end;
