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) 'or' All(b,PA,G)) '<' (a 'or' b)
proof
  let z be Element of Y;
A1: All(a,PA,G).z=TRUE or All(a,PA,G).z=FALSE by XBOOLEAN:def 3;
A2: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  assume (All(a,PA,G) 'or' All(b,PA,G)) .z=TRUE;
  then
A3: All(a,PA,G).z 'or' All(b,PA,G).z=TRUE by BVFUNC_1:def 4;
  per cases by A3,A1,BINARITH:3;
  suppose
A4: All(a,PA,G).z=TRUE;
A5: 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;
      hence contradiction by A4,BVFUNC_2:def 9;
    end;
    thus (a 'or' b).z = a.z 'or' b.z by BVFUNC_1:def 4
      .= TRUE 'or' b.z by A2,A5
      .= TRUE by BINARITH:10;
  end;
  suppose
A6: All(b,PA,G).z=TRUE;
A7: 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;
      hence contradiction by A6,BVFUNC_2:def 9;
    end;
    thus (a 'or' b).z = a.z 'or' b.z by BVFUNC_1:def 4
      .= a.z 'or' TRUE by A2,A7
      .= TRUE by BINARITH:10;
  end;
end;
