let Y be non empty set ; :: thesis: for G being Subset of ()
for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds
CompF (D,G) = ((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J

let G be Subset of (); :: thesis: for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds
CompF (D,G) = ((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J

let A, B, C, D, E, F, J be a_partition of Y; :: thesis: ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (D,G) = ((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J )
{A,B,C,D,E,F,J} = {A,B} \/ {C,D,E,F,J} by ENUMSET1:17
.= {A,B} \/ ({C,D} \/ {E,F,J}) by ENUMSET1:8
.= {A,B} \/ {D,C,E,F,J} by ENUMSET1:8
.= {A,B,D,C,E,F,J} by ENUMSET1:17 ;
hence ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (D,G) = ((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J ) by Th44; :: thesis: verum