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

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

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

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

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

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

{A,B,C,D} = {B,A,C,D} by ENUMSET1:65;

hence ( G = {A,B,C,D} & A <> B & B <> C & B <> D implies CompF (B,G) = (A '/\' C) '/\' D ) by Th7; :: thesis: verum

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

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

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

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

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

{A,B,C,D} = {B,A,C,D} by ENUMSET1:65;

hence ( G = {A,B,C,D} & A <> B & B <> C & B <> D implies CompF (B,G) = (A '/\' C) '/\' D ) by Th7; :: thesis: verum