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) '&' Ex(b,PA,G) '<' Ex(a '&' b,PA,G)
proof
  let z be Element of Y;
  assume (All(a,PA,G) '&' Ex(b,PA,G)) .z=TRUE;
  then
A1: All(a,PA,G).z '&' Ex(b,PA,G).z=TRUE by MARGREL1:def 20;
A2: now
    assume not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
    holds a.x=TRUE);
    then B_INF(a,CompF(PA,G)).z = FALSE by BVFUNC_1:def 16;
    then All(a,PA,G).z=FALSE by BVFUNC_2:def 9;
    hence contradiction by A1,MARGREL1:12;
  end;
  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;
    then Ex(b,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
A3: x1 in EqClass(z,CompF(PA,G)) and
A4: b.x1=TRUE;
  (a '&' b).x1 =a.x1 '&' b.x1 by MARGREL1:def 20
    .=TRUE '&' TRUE by A3,A4,A2
    .=TRUE;
  then B_SUP(a '&' b,CompF(PA,G)).z = TRUE by A3,BVFUNC_1:def 17;
  hence thesis by BVFUNC_2:def 10;
end;
