let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds
CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F
let G be Subset of (PARTITIONS Y); :: thesis: for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds
CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F
let A, B, C, D, E, F be a_partition of Y; :: thesis: ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F implies CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F )
A1: {A,B,C,D,E,F} = {A,B,C} \/ {D,E,F} by ENUMSET1:13
.= {A,B,C} \/ ({D,E} \/ {F}) by ENUMSET1:3
.= {A,B,C} \/ {E,D,F} by ENUMSET1:3
.= {A,B,C,E,D,F} by ENUMSET1:13 ;
assume ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) ; :: thesis: CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F
hence CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F by A1, Th34; :: thesis: verum