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 ENUMSET1:49;
then {A,B,C,D,E} = ({A} \/ {B,C}) \/ {D,E} by ENUMSET1:42;
then {A,B,C,D,E} = {A,C,B} \/ {D,E} by ENUMSET1:42;
then {A,B,C,D,E} = ({A,C} \/ {B}) \/ {D,E} by ENUMSET1:43;
then {A,B,C,D,E} = {C,A,B} \/ {D,E} by ENUMSET1:43;
then G = {C,A,B,D,E} by A1, ENUMSET1:49;
hence CompF C,G = ((A '/\' B) '/\' D) '/\' E by A2, Th24; :: thesis: verum