let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated steady-programmed definite realistic AMI-Struct of N
for il being Element of NAT
for i being Instruction of S st i is halting holds
NIC i,il = {il}
let S be non empty stored-program IC-Ins-separated steady-programmed definite realistic AMI-Struct of N; :: thesis: for il being Element of NAT
for i being Instruction of S st i is halting holds
NIC i,il = {il}
let il be Element of NAT ; :: thesis: for i being Instruction of S st i is halting holds
NIC i,il = {il}
let i be Instruction of S; :: thesis: ( i is halting implies NIC i,il = {il} )
assume A1: for s being State of S holds Exec i,s = s ; :: according to AMI_1:def 8 :: thesis: NIC i,il = {il}

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: {il} c= NIC i,il

let n be set ; :: thesis: ( n in NIC i,il implies n in {il} )
assume n in NIC i,il ; :: thesis: n in {il}
then ex s being Element of product the Object-Kind of S st
( n = IC (Following (ProgramPart s),s) & IC s = il & (ProgramPart s) /. il = i ) ;
then n = il by A1;
hence n in {il} by TARSKI:def 1; :: thesis: verum

end;

reconsider f = (IC S),il --> il,i as PartState of S by AMI_1:149;
consider s being State of S;
let n be set ; :: according to TARSKI:def 3 :: thesis: ( not n in {il} or n in NIC i,il )
assume A3: n in {il} ; :: thesis: n in NIC i,il
set a = s +* f;
A4: dom f = {(IC S),il} by FUNCT_4:65;
then IC S in dom f by TARSKI:def 2;
then A5: (s +* f) . (IC S) = f . (IC S) by FUNCT_4:14
.= il by AMI_1:48, FUNCT_4:66 ;
Y: il in dom f by A4, TARSKI:def 2;
(ProgramPart (s +* f)) /. (IC (s +* f)) = (s +* f) . (IC (s +* f)) by AMI_1:150
.= f . il by A5, FUNCT_4:14, Y
.= i by FUNCT_4:66 ;
then IC (Following (ProgramPart (s +* f)),(s +* f)) = (s +* f) . (IC S) by A1
.= n by A3, A5, TARSKI:def 1 ;
hence n in NIC i,il by Lm2; :: thesis: verum