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 <> C & B <> C & C <> D holds

CompF (C,G) = (A '/\' B) '/\' 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 <> C & B <> C & C <> D holds

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

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

{A,B,C,D} = {C,A,B,D} by ENUMSET1:67;

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

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

CompF (C,G) = (A '/\' B) '/\' 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 <> C & B <> C & C <> D holds

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

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

{A,B,C,D} = {C,A,B,D} by ENUMSET1:67;

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