reserve Y for non empty set,
  G for Subset of PARTITIONS(Y),
  A,B,C,D,E,F for a_partition of Y;

theorem Th34:
  G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D
& B<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F implies CompF(D,G) = A
  '/\' B '/\' C '/\' E '/\' F
proof
A1: {A,B,C,D,E,F} ={A,B} \/ {C,D,E,F} by ENUMSET1:12
    .={A,B} \/ ({C,D} \/ {E,F}) by ENUMSET1:5
    .={A,B} \/ {D,C,E,F} by ENUMSET1:5
    .={A,B,D,C,E,F} by ENUMSET1:12;
  assume G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B
  <>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F;
  hence thesis by A1,Th33;
end;
