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

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

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

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

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

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

assume that

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

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

{A,B,C,D,E,F} = {B,A} \/ {C,D,E,F} by ENUMSET1:12;

then G = {B,A,C,D,E,F} by A1, ENUMSET1:12;

hence CompF (B,G) = (((A '/\' C) '/\' D) '/\' E) '/\' F by A2, Th31; :: thesis: verum

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

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

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

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

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

assume that

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

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

{A,B,C,D,E,F} = {B,A} \/ {C,D,E,F} by ENUMSET1:12;

then G = {B,A,C,D,E,F} by A1, ENUMSET1:12;

hence CompF (B,G) = (((A '/\' C) '/\' D) '/\' E) '/\' F by A2, Th31; :: thesis: verum