let N be with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated steady-programmed definite standard AMI-Struct of NAT ,N
for F being programmed FinPartState of S st F is really-closed & il. S,0 in dom F holds
F is para-closed
let S be non empty stored-program IC-Ins-separated steady-programmed definite standard AMI-Struct of NAT ,N; :: thesis: for F being programmed FinPartState of S st F is really-closed & il. S,0 in dom F holds
F is para-closed
let F be programmed FinPartState of S; :: thesis: ( F is really-closed & il. S,0 in dom F implies F is para-closed )
assume that
A1: for s being State of S st F c= s & IC s in dom F holds
for k being Element of NAT holds IC (Computation s,k) in dom F and
A2: il. S,0 in dom F ; :: according to AMISTD_1:def 18 :: thesis: F is para-closed
let s be State of S; :: according to AMISTD_1:def 19 :: thesis: ( F c= s & IC s = il. S,0 implies for k being Element of NAT holds IC (Computation s,k) in dom F )
assume ( F c= s & IC s = il. S,0 ) ; :: thesis: for k being Element of NAT holds IC (Computation s,k) in dom F
hence for k being Element of NAT holds IC (Computation s,k) in dom F by A1, A2; :: thesis: verum