reserve Y for non empty set,
  G for Subset of PARTITIONS(Y),
  A,B,C,D,E,F for a_partition of Y;
reserve Y for non empty set,
  G for Subset of PARTITIONS(Y),
  A, B, C, D, E, F, J, M for a_partition of Y,
  x,x1,x2,x3,x4,x5,x6,x7,x8,x9 for set;

theorem Th70:
  for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y st G={A,B,C,D,E,F,J,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F &
A<>J & A<>M & A<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C
<>E & C<>F & C<>J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>
J & E<>M & E<>N & F<>J & F<>M & F<>N & J<>M & J<>N & M<>N holds CompF(D,G) = A
  '/\' B '/\' C '/\' E '/\' F '/\' J '/\' M '/\' N
proof
  let G be Subset of PARTITIONS(Y);
  let A,B,C,D,E,F,J,M,N be a_partition of Y;
  {A,B,C,D,E,F,J,M,N} ={A,B} \/ {C,D,E,F,J,M,N} by ENUMSET1:78
    .={A,B} \/ ({C,D} \/ {E,F,J,M,N}) by ENUMSET1:17
    .={A,B} \/ {D,C,E,F,J,M,N} by ENUMSET1:17
    .={A,B,D,C,E,F,J,M,N} by ENUMSET1:78;
  hence thesis by Th69;
end;
