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

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

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

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

assume that

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

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

{A,B,C,D,E} = {A,B,C} \/ {D,E} by ENUMSET1:9;

then {A,B,C,D,E} = ({A} \/ {B,C}) \/ {D,E} by ENUMSET1:2;

then {A,B,C,D,E} = {A,C,B} \/ {D,E} by ENUMSET1:2;

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

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

then G = {C,A,B,D,E} by A1, ENUMSET1:9;

hence CompF (C,G) = ((A '/\' B) '/\' D) '/\' E by A2, Th21; :: thesis: verum

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

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

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

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

assume that

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

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

{A,B,C,D,E} = {A,B,C} \/ {D,E} by ENUMSET1:9;

then {A,B,C,D,E} = ({A} \/ {B,C}) \/ {D,E} by ENUMSET1:2;

then {A,B,C,D,E} = {A,C,B} \/ {D,E} by ENUMSET1:2;

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

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

then G = {C,A,B,D,E} by A1, ENUMSET1:9;

hence CompF (C,G) = ((A '/\' B) '/\' D) '/\' E by A2, Th21; :: thesis: verum