let N be non empty with_zero set ; :: thesis: for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N
for F being Instruction-Sequence of S
for s being State of S
for k being 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 with_non-empty_values IC-Ins-separated halting AMI-Struct over N; :: thesis: for F being Instruction-Sequence of S
for s being State of S
for k being 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 Instruction-Sequence of S; :: thesis: for s being State of S
for k being 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 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 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
A3: dom F = NAT by PARTFUN1:def 2;

now :: thesis: not F . (IC (Comput (F,s,k))) = halt S

assume F . (IC (Comput (F,s,k))) = halt S ; :: thesis: contradiction
then CurInstr (F,(Comput (F,s,k))) = halt S by A3, PARTFUN1:def 6;
hence contradiction by A1, Th5, NAT_1:11; :: thesis: verum

end;

hence LifeSpan (F,s) = k + 1 by A2, Th31; :: thesis: verum