let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for 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 (F,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M) '/\' N
let G be Subset of (PARTITIONS Y); :: thesis: for 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 (F,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M) '/\' N
let A, B, C, D, E, F, J, M, N be a_partition of Y; :: thesis: ( 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 implies CompF (F,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M) '/\' N )
{A,B,C,D,E,F,J,M,N} = {A,B,C,D} \/ {E,F,J,M,N} by Lm3
.= {A,B,C,D} \/ ({E,F} \/ {J,M,N}) by ENUMSET1:8
.= {A,B,C,D} \/ {F,E,J,M,N} by ENUMSET1:8
.= {A,B,C,D,F,E,J,M,N} by Lm3 ;
hence ( 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 implies CompF (F,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M) '/\' N ) by Th71; :: thesis: verum