let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)

for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> E & B <> E & C <> E & D <> E holds

CompF (E,G) = ((A '/\' B) '/\' C) '/\' D

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

CompF (E,G) = ((A '/\' B) '/\' C) '/\' D

let A, B, C, D, E be a_partition of Y; :: thesis: ( G = {A,B,C,D,E} & A <> E & B <> E & C <> E & D <> E implies CompF (E,G) = ((A '/\' B) '/\' C) '/\' D )

assume that

A1: G = {A,B,C,D,E} and

A2: ( A <> E & B <> E & C <> E & D <> E ) ; :: thesis: CompF (E,G) = ((A '/\' B) '/\' C) '/\' D

{A,B,C,D,E} = {A,B,C} \/ {D,E} by ENUMSET1:9;

then G = {A,B,C,E,D} by A1, ENUMSET1:9;

hence CompF (E,G) = ((A '/\' B) '/\' C) '/\' D by A2, Th24; :: thesis: verum

for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> E & B <> E & C <> E & D <> E holds

CompF (E,G) = ((A '/\' B) '/\' C) '/\' D

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

CompF (E,G) = ((A '/\' B) '/\' C) '/\' D

let A, B, C, D, E be a_partition of Y; :: thesis: ( G = {A,B,C,D,E} & A <> E & B <> E & C <> E & D <> E implies CompF (E,G) = ((A '/\' B) '/\' C) '/\' D )

assume that

A1: G = {A,B,C,D,E} and

A2: ( A <> E & B <> E & C <> E & D <> E ) ; :: thesis: CompF (E,G) = ((A '/\' B) '/\' C) '/\' D

{A,B,C,D,E} = {A,B,C} \/ {D,E} by ENUMSET1:9;

then G = {A,B,C,E,D} by A1, ENUMSET1:9;

hence CompF (E,G) = ((A '/\' B) '/\' C) '/\' D by A2, Th24; :: thesis: verum