reserve Y for non empty set;

theorem
  for a,b being Function of Y,BOOLEAN, G being Subset of
PARTITIONS(Y), PA being a_partition of Y holds All(a '&' b,PA,G) '<' a '&' All(
  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
A1: All(a '&' b,PA,G).z=TRUE;
A2: now
    assume not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
    holds a.x=TRUE);
    then consider x1 being Element of Y such that
A3: x1 in EqClass(z,CompF(PA,G)) and
A4: a.x1<>TRUE;
    (a '&' b).x1=TRUE by A1,A3,BVFUNC_1:def 16;
    then
A5: a.x1 '&' b.x1=TRUE by MARGREL1:def 20;
    a.x1=FALSE by A4,XBOOLEAN:def 3;
    hence contradiction by A5;
  end;
  now
    assume not (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds
    b.x=TRUE);
    then consider x1 being Element of Y such that
A6: x1 in EqClass(z,CompF(PA,G)) and
A7: b.x1<>TRUE;
    (a '&' b).x1=TRUE by A1,A6,BVFUNC_1:def 16;
    then
A8: a.x1 '&' b.x1=TRUE by MARGREL1:def 20;
    b.x1=FALSE by A7,XBOOLEAN:def 3;
    hence contradiction by A8;
  end;
  then
A9: B_INF(b,CompF(PA,G)).z = TRUE by BVFUNC_1:def 16;
  z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  then a.z=TRUE by A2;
  then (a '&' All(b,PA,G)).z =TRUE '&' TRUE by A9,MARGREL1:def 20
    .=TRUE;
  hence thesis;
end;
