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

theorem
  for a,b being Function of Y,BOOLEAN, PA being a_partition of Y
  holds All(a,PA,G) 'or' All(b,PA,G) '<' All(a 'or' b,PA,G)
proof
  let a,b be Function of Y,BOOLEAN;
  let PA be a_partition of Y;
  let z be Element of Y;
  assume (All(a,PA,G) 'or' All(b,PA,G)).z=TRUE;
  then
A1: All(a,PA,G).z 'or' All(b,PA,G).z=TRUE by BVFUNC_1:def 4;
A2: All(b,PA,G).z=TRUE or All(b,PA,G).z=FALSE by XBOOLEAN:def 3;
  now
    per cases by A1,A2,BINARITH:3;
    case
A3:   All(a,PA,G).z=TRUE;
      for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds (a
      'or' b).x=TRUE
      proof
        let x be Element of Y;
        assume
A4:     x in EqClass(z,CompF(PA,G));
        (a 'or' b).x =a.x 'or' b.x by BVFUNC_1:def 4
          .=TRUE 'or' b.x by A3,A4,BVFUNC_1:def 16
          .=TRUE by BINARITH:10;
        hence thesis;
      end;
      hence thesis by BVFUNC_1:def 16;
    end;
    case
A5:   All(b,PA,G).z=TRUE;
      for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds (a
      'or' b).x=TRUE
      proof
        let x be Element of Y;
        assume
A6:     x in EqClass(z,CompF(PA,G));
        (a 'or' b).x =a.x 'or' b.x by BVFUNC_1:def 4
          .=a.x 'or' TRUE by A5,A6,BVFUNC_1:def 16
          .=TRUE by BINARITH:10;
        hence thesis;
      end;
      hence thesis by BVFUNC_1:def 16;
    end;
  end;
  hence thesis;
end;
