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

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

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

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

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

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

{A,B,C} = {C,A,B} by ENUMSET1:59;

hence ( G = {A,B,C} & B <> C & C <> A implies CompF (C,G) = A '/\' B ) by Th4; :: thesis: verum

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

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

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

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

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

{A,B,C} = {C,A,B} by ENUMSET1:59;

hence ( G = {A,B,C} & B <> C & C <> A implies CompF (C,G) = A '/\' B ) by Th4; :: thesis: verum