let s be State of ; :: thesis: for I being Program of
for a being Int-Location
for k being Element of NAT st I is_closed_on Initialize s & I is_halting_on Initialize s & I does_not_destroy a holds
(IExec I,s) . a = (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . a
set A = NAT ;
let I be Program of ; :: thesis: for a being Int-Location
for k being Element of NAT st I is_closed_on Initialize s & I is_halting_on Initialize s & I does_not_destroy a holds
(IExec I,s) . a = (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . a
let a be Int-Location ; :: thesis: for k being Element of NAT st I is_closed_on Initialize s & I is_halting_on Initialize s & I does_not_destroy a holds
(IExec I,s) . a = (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . a
let k be Element of NAT ; :: thesis: ( I is_closed_on Initialize s & I is_halting_on Initialize s & I does_not_destroy a implies (IExec I,s) . a = (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . a )
assume A1: I is_closed_on Initialize s ; :: thesis: ( not I is_halting_on Initialize s or not I does_not_destroy a or (IExec I,s) . a = (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . a )

now

assume A2: a in NAT ; :: thesis: contradiction
then reconsider a = a as Instruction-Location of SCM+FSA by AMI_1:def 4;
a in NAT by A2;
hence contradiction by SCMFSA_2:84; :: thesis: verum

end;

then not a in (dom s) /\ NAT by XBOOLE_0:def 4;
then A3: not a in dom (s | NAT ) by RELAT_1:90;
set s0 = Initialize s;
set s1 = (Initialize s) +* (I +* (Start-At (insloc 0 )));
assume I is_halting_on Initialize s ; :: thesis: ( not I does_not_destroy a or (IExec I,s) . a = (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . a )
then A4: ProgramPart ((Initialize s) +* (I +* (Start-At (insloc 0 )))) halts_on (Initialize s) +* (I +* (Start-At (insloc 0 ))) by SCMFSA7B:def 8;
assume A5: I does_not_destroy a ; :: thesis: (IExec I,s) . a = (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . a
s +* (Initialized I) = (Initialize s) +* (I +* (Start-At (insloc 0 ))) by SCMFSA8A:13;
hence (IExec I,s) . a = ((Result ((Initialize s) +* (I +* (Start-At (insloc 0 ))))) +* (s | NAT )) . a by SCMFSA6B:def 1
.= (Result ((Initialize s) +* (I +* (Start-At (insloc 0 ))))) . a by A3, FUNCT_4:12
.= (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),(LifeSpan ((Initialize s) +* (I +* (Start-At (insloc 0 )))))) . a by A4, AMI_1:122
.= (Initialize s) . a by A1, A5, SCMFSA7B:27
.= (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . a by A1, A5, SCMFSA7B:27 ;
:: thesis: verum