let Y be non empty set ; :: thesis: for G being Subset of ()
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 (E,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M

let G be Subset of (); :: thesis: 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 (E,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M

let A, B, C, D, E, F, J, M be a_partition of Y; :: thesis: ( 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 (E,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' 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 ENUMSET1:8
.= {A,B,C} \/ {E,D,F,J,M} by ENUMSET1:8
.= {A,B,C,E,D,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 (E,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M ) by Th57; :: thesis: verum