let Y be non empty set ; for G being Subset of (PARTITIONS Y)
for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds
CompF (C,G) = (((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M
let G be Subset of (PARTITIONS Y); for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds
CompF (C,G) = (((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M
let A, B, C, D, E, F, J, M be a_partition of Y; ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (C,G) = (((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M )
{A,B,C,D,E,F,J,M} =
{A,B,C} \/ {D,E,F,J,M}
by ENUMSET1:24
.=
({A} \/ {B,C}) \/ {D,E,F,J,M}
by ENUMSET1:2
.=
{A,C,B} \/ {D,E,F,J,M}
by ENUMSET1:2
.=
{A,C,B,D,E,F,J,M}
by ENUMSET1:24
;
hence
( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (C,G) = (((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M )
by Th55; verum