let s be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA st I is_halting_on Initialized s holds
( ( for a being read-write Int-Location holds (IExec I,s) . a = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . a ) & ( for f being FinSeq-Location holds (IExec I,s) . f = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . f ) )
set A = NAT ;
let I be Program of SCM+FSA ; :: thesis: ( I is_halting_on Initialized s implies ( ( for a being read-write Int-Location holds (IExec I,s) . a = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . a ) & ( for f being FinSeq-Location holds (IExec I,s) . f = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . f ) ) )
set s0 = Initialized s;
set s1 = (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ));
assume I is_halting_on Initialized s ; :: thesis: ( ( for a being read-write Int-Location holds (IExec I,s) . a = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . a ) & ( for f being FinSeq-Location holds (IExec I,s) . f = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . f ) )
then A1: ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))) halts_on (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )) by SCMFSA7B:def 8;

hereby :: thesis: for f being FinSeq-Location holds (IExec I,s) . f = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . f

let a be read-write Int-Location ; :: thesis: (IExec I,s) . a = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . a
not a in NAT by SCMFSA_2:84;
then not a in (dom s) /\ NAT by XBOOLE_0:def 4;
then A3: not a in dom (s | NAT ) by RELAT_1:90;
s +* (Initialized I) = (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )) by SCMFSA8A:13;
hence (IExec I,s) . a = ((Result (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))) +* (s | NAT )) . a by SCMFSA6B:def 1
.= (Result (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))) . a by A3, FUNCT_4:12
.= (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . a by A1, AMI_1:122 ;
:: thesis: verum

end;

let f be FinSeq-Location ; :: thesis: (IExec I,s) . f = (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . f
not f in NAT by SCMFSA_2:85;
then not f in (dom s) /\ NAT by XBOOLE_0:def 4;
then A5: not f in dom (s | NAT ) by RELAT_1:90;
s +* (Initialized I) = (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )) by SCMFSA8A:13;
hence (IExec I,s) . f = ((Result (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))) +* (s | NAT )) . f by SCMFSA6B:def 1
.= (Result (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))) . f by A5, FUNCT_4:12
.= (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) . f by A1, AMI_1:122 ;
:: thesis: verum