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' Ex(a '&' b,PA,G) '<' 'not' All(a 'imp' b,PA,G)
proof
  let z be Element of Y;
  assume (Ex(a,PA,G) '&' 'not' Ex(a '&' b,PA,G)).z=TRUE;
  then
A1: Ex(a,PA,G).z '&' ('not' Ex(a '&' b,PA,G)).z=TRUE by MARGREL1:def 20;
  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;
    then Ex(a,PA,G).z=FALSE by BVFUNC_2:def 10;
    hence contradiction by A1,MARGREL1:12;
  end;
  then consider x1 being Element of Y such that
A2: x1 in EqClass(z,CompF(PA,G)) and
A3: a.x1=TRUE;
  ('not' Ex(a '&' b,PA,G)).z=TRUE by A1,MARGREL1:12;
  then 'not' Ex(a '&' b,PA,G).z=TRUE by MARGREL1:def 19;
  then
  Ex(a '&' b,PA,G) = B_SUP(a '&' b,CompF(PA,G)) & Ex(a '&' b,PA,G).z=FALSE
  by BVFUNC_2:def 10,MARGREL1:11;
  then (a '&' b).x1<>TRUE by A2,BVFUNC_1:def 17;
  then (a '&' b).x1=FALSE by XBOOLEAN:def 3;
  then
A4: a.x1 '&' b.x1=FALSE by MARGREL1:def 20;
  per cases by A4,MARGREL1:12;
  suppose
    a.x1=FALSE;
    hence thesis by A3;
  end;
  suppose
    b.x1=FALSE;
    then (a 'imp' b).x1 =('not' TRUE) 'or' FALSE by A3,BVFUNC_1:def 8
      .=FALSE 'or' FALSE by MARGREL1:11
      .=FALSE;
    then B_INF(a 'imp' b,CompF(PA,G)).z = FALSE by A2,BVFUNC_1:def 16;
    then All(a 'imp' b,PA,G).z=FALSE by BVFUNC_2:def 9;
    hence ('not' All(a 'imp' b,PA,G)).z ='not' FALSE by MARGREL1:def 19
      .=TRUE by MARGREL1:11;
  end;
end;
