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

theorem Th23:
  G={A,B,C,D,E} & A<>C & B<>C & C<>D & C<>E implies CompF(C,G) = A
  '/\' B '/\' D '/\' E
proof
  assume that
A1: G={A,B,C,D,E} and
A2: A<>C & B<>C & C<>D & C<>E;
  {A,B,C,D,E}={A,B,C} \/ {D,E} by ENUMSET1:9;
  then {A,B,C,D,E}={A} \/ {B,C} \/ {D,E} by ENUMSET1:2;
  then {A,B,C,D,E}={A,C,B} \/ {D,E} by ENUMSET1:2;
  then {A,B,C,D,E}={A,C} \/ {B} \/ {D,E} by ENUMSET1:3;
  then {A,B,C,D,E}={C,A,B} \/ {D,E} by ENUMSET1:3;
  then G={C,A,B,D,E} by A1,ENUMSET1:9;
  hence thesis by A2,Th21;
end;
