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 st ex k being Nat st F . (IC (Comput (F,s,k))) = halt S holds
F halts_on s

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 st ex k being Nat st F . (IC (Comput (F,s,k))) = halt S holds
F halts_on s

let F be Instruction-Sequence of S; :: thesis: for s being State of S st ex k being Nat st F . (IC (Comput (F,s,k))) = halt S holds
F halts_on s

let s be State of S; :: thesis: ( ex k being Nat st F . (IC (Comput (F,s,k))) = halt S implies F halts_on s )
given k being Nat such that A1: F . (IC (Comput (F,s,k))) = halt S ; :: thesis: F halts_on s
take k ; :: according to EXTPRO_1:def 8 :: thesis: ( IC (Comput (F,s,k)) in dom F & CurInstr (F,(Comput (F,s,k))) = halt S )
A2: dom F = NAT by PARTFUN1:def 2;
hence IC (Comput (F,s,k)) in dom F ; :: thesis: CurInstr (F,(Comput (F,s,k))) = halt S
thus CurInstr (F,(Comput (F,s,k))) = halt S by ; :: thesis: verum