let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty IC-Ins-separated weakly_standard AMI-Struct of N
for F being NAT -defined the Instructions of b₁ -valued finite Function st F is really-closed & il. (S,0) in dom F holds
F is para-closed
let S be non empty IC-Ins-separated weakly_standard AMI-Struct of N; :: thesis: for F being NAT -defined the Instructions of S -valued finite Function st F is really-closed & il. (S,0) in dom F holds
F is para-closed
let F be NAT -defined the Instructions of S -valued finite Function; :: thesis: ( F is really-closed & il. (S,0) in dom F implies F is para-closed )
assume F is really-closed ; :: thesis: ( not il. (S,0) in dom F or F is para-closed )
then A1: for s being State of S st IC s in dom F holds
for k being Element of NAT holds IC (Comput (F,s,k)) in dom F by AMISTD_1:14;
assume B1: il. (S,0) in dom F ; :: thesis: F is para-closed
let s be State of S; :: according to AMI_WSTD:def 9 :: thesis: ( IC s = il. (S,0) implies for k being Element of NAT holds IC (Comput (F,s,k)) in dom F )
assume IC s = il. (S,0) ; :: thesis: for k being Element of NAT holds IC (Comput (F,s,k)) in dom F
hence for k being Element of NAT holds IC (Comput (F,s,k)) in dom F by A1, B1; :: thesis: verum