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) '&' All(b,PA,G)) '<' (a '&' b)
proof
  let z be Element of Y;
A1: (All(a,PA,G) '&' All(b,PA,G)).z =All(a,PA,G).z '&' All(b,PA,G).z by
MARGREL1:def 20;
  assume
A2: (All(a,PA,G) '&' All(b,PA,G)).z=TRUE;
A3: 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 A2,A1,MARGREL1:12;
  end;
A4: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  now
    assume not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
    holds b.x=TRUE);
    then B_INF(b,CompF(PA,G)).z = FALSE by BVFUNC_1:def 16;
    then All(b,PA,G).z=FALSE by BVFUNC_2:def 9;
    hence contradiction by A2,A1,MARGREL1:12;
  end;
  then
A5: b.z=TRUE by A4;
  thus (a '&' b).z =a.z '&' b.z by MARGREL1:def 20
    .=TRUE '&' TRUE by A4,A3,A5
    .=TRUE;
end;
