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

theorem Th32:
  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(B,G) = A
  '/\' C '/\' D '/\' E '/\' F
proof
  assume that
A1: G={A,B,C,D,E,F} and
A2: 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;
  {A,B,C,D,E,F}={B,A} \/ {C,D,E,F} by ENUMSET1:12;
  then G={B,A,C,D,E,F} by A1,ENUMSET1:12;
  hence thesis by A2,Th31;
end;
