let A, B, C, D, E, F, J, M, N be set ; :: thesis: for h being Function
for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds
dom h = {A,B,C,D,E,F,J,M,N}
let h be Function; :: thesis: for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds
dom h = {A,B,C,D,E,F,J,M,N}
let A9, B9, C9, D9, E9, F9, J9, M9, N9 be set ; :: thesis: ( h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) implies dom h = {A,B,C,D,E,F,J,M,N} )
assume A1: h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) ; :: thesis: dom h = {A,B,C,D,E,F,J,M,N}
A2: dom (A .--> A9) = {A} by FUNCOP_1:19;
dom ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) = {N,B,C,D,E,F,J,M} by Th66
.= {N} \/ {B,C,D,E,F,J,M} by ENUMSET1:62
.= {B,C,D,E,F,J,M,N} by ENUMSET1:68 ;
then dom (((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9)) = {B,C,D,E,F,J,M,N} \/ {A} by A2, FUNCT_4:def 1
.= {A,B,C,D,E,F,J,M,N} by ENUMSET1:127 ;
hence dom h = {A,B,C,D,E,F,J,M,N} by A1; :: thesis: verum