let N be 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 st ex k being Element of NAT st s . (IC (Computation s,k)) = halt S holds
for i being Element of NAT holds Result s = Result (Computation s,i)
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 st ex k being Element of NAT st s . (IC (Computation s,k)) = halt S holds
for i being Element of NAT holds Result s = Result (Computation s,i)
let s be State of S; :: thesis: ( ex k being Element of NAT st s . (IC (Computation s,k)) = halt S implies for i being Element of NAT holds Result s = Result (Computation s,i) )
given k being Element of NAT such that A1: s . (IC (Computation s,k)) = halt S ; :: thesis: for i being Element of NAT holds Result s = Result (Computation s,i)
let i be Element of NAT ; :: thesis: Result s = Result (Computation s,i)
set s' = Computation s,k;
A2: CurInstr (Computation s,k) = halt S by A1, Th54;
IC (Computation s,k) in NAT by Def4;
then X: IC (Computation s,k) in dom (ProgramPart (Computation s,i)) by LmU;

now

per cases ( i <= k or i >= k ) ;

suppose i <= k ; :: thesis: Result s = Result (Computation s,i)

then consider j being Nat such that
A3: k = i + j by NAT_1:10;
reconsider j = j as Element of NAT by ORDINAL1:def 13;
A4: Computation s,k = Computation (Computation s,i),j by A3, Th51;
then (ProgramPart (Computation s,i)) . (IC (Computation s,k)) = halt S by A2, LmX;
then A5: ProgramPart (Computation s,i) halts_on Computation s,i by A4, Def20, X;
thus Result s = Computation s,k by A1, Th56
.= Result (Computation s,i) by A2, A4, A5, Def22 ; :: thesis: verum

end;

suppose A6: i >= k ; :: thesis: Result s = Result (Computation s,i)

A7: Computation (Computation s,k),0 = Computation s,k by Th13;
A8: Computation s,i = Computation s,k by A2, A6, Th52;
IC (Computation (Computation s,k),0 ) in NAT by Def4;
then X: IC (Computation (Computation s,k),0 ) in dom (ProgramPart (Computation s,k)) by LmU;
(ProgramPart (Computation s,k)) . (IC (Computation (Computation s,k),0 )) = halt S by A2, A7, LmX;
then A9: ProgramPart (Computation s,i) halts_on Computation s,i by Def20, X, A8;
thus Result s = Computation s,k by A1, Th56
.= Result (Computation s,i) by A2, A8, A7, A9, Def22 ; :: thesis: verum

end;

end;

end;

hence Result s = Result (Computation s,i) ; :: thesis: verum