let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty IC-Ins-separated halting AMI-Struct of N
for l being Element of NAT
for P being NAT -defined the Instructions of b₁ -valued Function st l .--> (halt S) c= P holds
for p being b₂ -started PartState of S
for s being State of S st p c= s holds
for i being Element of NAT holds Comput (P,s,i) = s
let S be non empty IC-Ins-separated halting AMI-Struct of N; :: thesis: for l being Element of NAT
for P being NAT -defined the Instructions of S -valued Function st l .--> (halt S) c= P holds
for p being b₁ -started PartState of S
for s being State of S st p c= s holds
for i being Element of NAT holds Comput (P,s,i) = s
let l be Element of NAT ; :: thesis: for P being NAT -defined the Instructions of S -valued Function st l .--> (halt S) c= P holds
for p being l -started PartState of S
for s being State of S st p c= s holds
for i being Element of NAT holds Comput (P,s,i) = s
let P be NAT -defined the Instructions of S -valued Function; :: thesis: ( l .--> (halt S) c= P implies for p being l -started PartState of S
for s being State of S st p c= s holds
for i being Element of NAT holds Comput (P,s,i) = s )
assume A1: l .--> (halt S) c= P ; :: thesis: for p being l -started PartState of S
for s being State of S st p c= s holds
for i being Element of NAT holds Comput (P,s,i) = s
let p be l -started PartState of S; :: thesis: for s being State of S st p c= s holds
for i being Element of NAT holds Comput (P,s,i) = s
set h = halt S;
let s be State of S; :: thesis: ( p c= s implies for i being Element of NAT holds Comput (P,s,i) = s )
assume A2: p c= s ; :: thesis: for i being Element of NAT holds Comput (P,s,i) = s
A3: Start-At (l,S) c= p by MEMSTR_0:29;
defpred S₁[ Element of NAT ] means Comput (P,s,$1) = s;
A4: Start-At (l,S) c= s by A3, A2, XBOOLE_1:1;

A5: now

let i be Element of NAT ; :: thesis: ( S₁[i] implies S₁[i + 1] )
assume A6: S₁[i] ; :: thesis: S₁[i + 1]
Comput (P,s,(i + 1)) = Following (P,s) by A6, Th4
.= Exec ((halt S),s) by A4, A1, Th10
.= s by Def3 ;
hence S₁[i + 1] ; :: thesis: verum

end;

A7: S₁[ 0 ] by Th3;
thus for i being Element of NAT holds S₁[i] from NAT_1:sch 1(A7, A5); :: thesis: verum