let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite halting AMI-Struct of N
for F being NAT -defined the Instructions of b₁ -valued total Function
for s being State of S
for k being Element of NAT st IC (Comput (F,s,k)) <> IC (Comput (F,s,(k + 1))) & F . (IC (Comput (F,s,(k + 1)))) = halt S holds
LifeSpan (F,s) = k + 1
let S be non empty stored-program IC-Ins-separated definite halting AMI-Struct of N; :: thesis: for F being NAT -defined the Instructions of S -valued total Function
for s being State of S
for k being Element of NAT st IC (Comput (F,s,k)) <> IC (Comput (F,s,(k + 1))) & F . (IC (Comput (F,s,(k + 1)))) = halt S holds
LifeSpan (F,s) = k + 1
let F be NAT -defined the Instructions of S -valued total Function; :: thesis: for s being State of S
for k being Element of NAT st IC (Comput (F,s,k)) <> IC (Comput (F,s,(k + 1))) & F . (IC (Comput (F,s,(k + 1)))) = halt S holds
LifeSpan (F,s) = k + 1
let s be State of S; :: thesis: for k being Element of NAT st IC (Comput (F,s,k)) <> IC (Comput (F,s,(k + 1))) & F . (IC (Comput (F,s,(k + 1)))) = halt S holds
LifeSpan (F,s) = k + 1
let k be Element of NAT ; :: thesis: ( IC (Comput (F,s,k)) <> IC (Comput (F,s,(k + 1))) & F . (IC (Comput (F,s,(k + 1)))) = halt S implies LifeSpan (F,s) = k + 1 )
assume that
A1: IC (Comput (F,s,k)) <> IC (Comput (F,s,(k + 1))) and
A2: F . (IC (Comput (F,s,(k + 1)))) = halt S ; :: thesis: LifeSpan (F,s) = k + 1
dom F = NAT by PARTFUN1:def 4;
then XX: IC (Comput (F,s,k)) in dom F ;

now

assume F . (IC (Comput (F,s,k))) = halt S ; :: thesis: contradiction
then CurInstr (F,(Comput (F,s,k))) = halt S by PARTFUN1:def 8, XX;
hence contradiction by A1, Th6, NAT_1:11; :: thesis: verum

end;

hence LifeSpan (F,s) = k + 1 by A2, Th33; :: thesis: verum