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 st ex k being Element of NAT st F . (IC (Comput (F,s,k))) = halt S holds
F halts_on s
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 st ex k being Element of NAT st F . (IC (Comput (F,s,k))) = halt S holds
F halts_on s
let F be NAT -defined the Instructions of S -valued total Function; :: thesis: for s being State of S st ex k being Element of 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 Element of NAT st F . (IC (Comput (F,s,k))) = halt S implies F halts_on s )
given k being Element of NAT such that A1: F . (IC (Comput (F,s,k))) = halt S ; :: thesis: F halts_on s
take k ; :: according to EXTPRO_1:def 7 :: thesis: ( IC (Comput (F,s,k)) in dom F & CurInstr (F,(Comput (F,s,k))) = halt S )
dom F = NAT by PARTFUN1:def 4;
hence IC (Comput (F,s,k)) in dom F ; :: thesis: CurInstr (F,(Comput (F,s,k))) = halt S
hence CurInstr (F,(Comput (F,s,k))) = halt S by A1, PARTFUN1:def 8; :: thesis: verum