let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of N
for s being State of S
for k being Element of NAT st CurInstr (ProgramPart (Comput (ProgramPart s),s,k)),(Comput (ProgramPart s),s,k) = halt S holds
Comput (ProgramPart s),s,(LifeSpan s) = Comput (ProgramPart s),s,k
let S be non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of N; :: thesis: for s being State of S
for k being Element of NAT st CurInstr (ProgramPart (Comput (ProgramPart s),s,k)),(Comput (ProgramPart s),s,k) = halt S holds
Comput (ProgramPart s),s,(LifeSpan s) = Comput (ProgramPart s),s,k
let s be State of S; :: thesis: for k being Element of NAT st CurInstr (ProgramPart (Comput (ProgramPart s),s,k)),(Comput (ProgramPart s),s,k) = halt S holds
Comput (ProgramPart s),s,(LifeSpan s) = Comput (ProgramPart s),s,k
let k be Element of NAT ; :: thesis: ( CurInstr (ProgramPart (Comput (ProgramPart s),s,k)),(Comput (ProgramPart s),s,k) = halt S implies Comput (ProgramPart s),s,(LifeSpan s) = Comput (ProgramPart s),s,k )
assume A1: CurInstr (ProgramPart (Comput (ProgramPart s),s,k)),(Comput (ProgramPart s),s,k) = halt S ; :: thesis: Comput (ProgramPart s),s,(LifeSpan s) = Comput (ProgramPart s),s,k
IC (Comput (ProgramPart s),s,k) in NAT ;
then X: IC (Comput (ProgramPart s),s,k) in dom (ProgramPart s) by LmU;
(ProgramPart s) /. (IC (Comput (ProgramPart s),s,k)) = CurInstr (ProgramPart (Comput (ProgramPart s),s,k)),(Comput (ProgramPart s),s,k) by LmX;
then A2: ProgramPart s halts_on s by Def20, X, A1;
set Ls = LifeSpan s;
A3: CurInstr (ProgramPart (Comput (ProgramPart s),s,(LifeSpan s))),(Comput (ProgramPart s),s,(LifeSpan s)) = halt S by A2, Def46;
X: ProgramPart (Comput (ProgramPart s),s,(LifeSpan s)) = ProgramPart s by LmY;
LifeSpan s <= k by A1, A2, Def46;
hence Comput (ProgramPart s),s,(LifeSpan s) = Comput (ProgramPart s),s,k by A3, Th52, X; :: thesis: verum