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
  (a '&' b) '<' (Ex(a,PA,G) '&' Ex(b,PA,G))
proof
  let z be Element of Y;
A1: (a '&' b).z=a.z '&' b.z by MARGREL1:def 20;
  assume
A2: (a '&' b).z=TRUE;
  then
A3: Ex(a,PA,G) = B_SUP(a,CompF(PA,G)) & a.z=TRUE by A1,BVFUNC_2:def 10
,MARGREL1:12;
A4: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  b.z=TRUE by A2,A1,MARGREL1:12;
  then B_SUP(b,CompF(PA,G)).z = TRUE by A4,BVFUNC_1:def 17;
  then
A5: Ex(b,PA,G).z=TRUE by BVFUNC_2:def 10;
  thus (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 A3,A4,A5,BVFUNC_1:def 17
    .=TRUE;
end;
