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 ENUMSET1:49;
then G = {A,B,C,E,D} by A1, ENUMSET1:49;
hence CompF E,G = ((A '/\' B) '/\' C) '/\' D by A2, Th27; :: thesis: verum