let N be with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated steady-programmed definite AMI-Struct of N
for F being NAT -defined FinPartState of st F is closed holds
F is really-closed
let S be non empty stored-program IC-Ins-separated steady-programmed definite AMI-Struct of N; :: thesis: for F being NAT -defined FinPartState of st F is closed holds
F is really-closed
let F be NAT -defined FinPartState of ; :: thesis: ( F is closed implies F is really-closed )
assume A1: F is closed ; :: thesis: F is really-closed
let s be State of ; :: according to AMISTD_1:def 18 :: thesis: ( F c= s & IC s in dom F implies for k being Element of NAT holds IC (Computation s,k) in dom F )
assume that
A2: F c= s and
A3: IC s in dom F ; :: thesis: for k being Element of NAT holds IC (Computation s,k) in dom F
defpred S₁[ Element of NAT ] means IC (Computation s,$1) in dom F;

A4: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: S₁[k] ; :: thesis: S₁[k + 1]
set t = Computation s,k;
set l = IC (Computation s,k);
( pi F,(IC (Computation s,k)) = F . (IC (Computation s,k)) & F . (IC (Computation s,k)) = s . (IC (Computation s,k)) ) by A2, A5, AMI_1:def 47, GRFUNC_1:8;
then (Computation s,k) . (IC (Computation s,k)) = pi F,(IC (Computation s,k)) by AMI_1:54;
then A6: IC (Following (Computation s,k)) in NIC (pi F,(IC (Computation s,k))),(IC (Computation s,k)) ;
A7: Computation s,(k + 1) = Following (Computation s,k) by AMI_1:14;
NIC (pi F,(IC (Computation s,k))),(IC (Computation s,k)) c= dom F by A1, A5, Def17;
hence S₁[k + 1] by A6, A7; :: thesis: verum

end;

A8: S₁[ 0 ] by A3, AMI_1:13;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A8, A4); :: thesis: verum