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

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

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

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

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

assume that

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

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

{A,B,C,D,E} = {A,B} \/ {C,D,E} by ENUMSET1:8;

then {A,B,C,D,E} = {A,B} \/ ({C,D} \/ {E}) by ENUMSET1:3;

then {A,B,C,D,E} = {A,B} \/ {D,C,E} by ENUMSET1:3;

then G = {A,B,D,C,E} by A1, ENUMSET1:8;

hence CompF (D,G) = ((A '/\' B) '/\' C) '/\' E by A2, Th23; :: thesis: verum

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

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

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

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

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

assume that

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

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

{A,B,C,D,E} = {A,B} \/ {C,D,E} by ENUMSET1:8;

then {A,B,C,D,E} = {A,B} \/ ({C,D} \/ {E}) by ENUMSET1:3;

then {A,B,C,D,E} = {A,B} \/ {D,C,E} by ENUMSET1:3;

then G = {A,B,D,C,E} by A1, ENUMSET1:8;

hence CompF (D,G) = ((A '/\' B) '/\' C) '/\' E by A2, Th23; :: thesis: verum