reserve Y for non empty set,
  G for Subset of PARTITIONS(Y);

theorem
  for a,b being Function of Y,BOOLEAN,G being Subset of PARTITIONS
(Y) , PA being a_partition of Y holds Ex(a '&' b,PA,G) '<' Ex(a,PA,G) '&' Ex(b,
  PA,G)
proof
  let a,b be Function of Y,BOOLEAN;
  let G be Subset of PARTITIONS(Y);
  let PA be a_partition of Y;
  let z be Element of Y;
  assume Ex(a '&' b,PA,G).z=TRUE;
  then consider x1 being Element of Y such that
A1: x1 in EqClass(z,CompF(PA,G)) and
A2: (a '&' b).x1=TRUE by BVFUNC_1:def 17;
A3: a.x1 '&' b.x1=TRUE by A2,MARGREL1:def 20;
  then
A4: b.x1=TRUE by MARGREL1:12;
  a.x1=TRUE by A3,MARGREL1:12;
  then
A5: Ex(a,PA,G).z=TRUE by A1,BVFUNC_1:def 17;
  (Ex(a,PA,G) '&' Ex(b,PA,G)).z = Ex(a,PA,G).z '&' Ex(b,PA,G).z by
MARGREL1:def 20
    .= TRUE '&' TRUE by A1,A4,A5,BVFUNC_1:def 17
    .= TRUE;
  hence thesis;
end;
