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

theorem Th24:
  G={A,B,C,D,E} & A<>D & B<>D & C<>D & D<>E implies CompF(D,G) = A
  '/\' B '/\' C '/\' E
proof
  assume that
A1: G={A,B,C,D,E} and
A2: A<>D & B<>D & C<>D & D<>E;
  {A,B,C,D,E}={A,B} \/ {C,D,E} by ENUMSET1:8;
  then {A,B,C,D,E}={A,B} \/ ({C,D} \/ {E}) by ENUMSET1:3;
  then {A,B,C,D,E}={A,B} \/ {D,C,E} by ENUMSET1:3;
  then G={A,B,D,C,E} by A1,ENUMSET1:8;
  hence thesis by A2,Th23;
end;
