let s be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA
for a being Int-Location st not I destroysdestroy a holds
for k being Element of NAT st IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) in dom I holds
(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1)) . a = (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a
let I be Program of SCM+FSA ; :: thesis: for a being Int-Location st not I destroysdestroy a holds
for k being Element of NAT st IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) in dom I holds
(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1)) . a = (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a
let a be Int-Location ; :: thesis: ( not I destroysdestroy a implies for k being Element of NAT st IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) in dom I holds
(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1)) . a = (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a )
assume A1: not I destroysdestroy a ; :: thesis: for k being Element of NAT st IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) in dom I holds
(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1)) . a = (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a
set s1 = s +* (I +* (Start-At 0 ,SCM+FSA ));
let k be Element of NAT ; :: thesis: ( IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) in dom I implies (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1)) . a = (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a )
assume A2: IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) in dom I ; :: thesis: (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1)) . a = (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a
dom I misses dom (Start-At 0 ,SCM+FSA ) by SF_MASTR:64;
then A3: I c= I +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:33;
set l = IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k);
I +* (Start-At 0 ,SCM+FSA ) c= s +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
then I c= s +* (I +* (Start-At 0 ,SCM+FSA )) by A3, XBOOLE_1:1;
then (s +* (I +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) = I . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) by A2, GRFUNC_1:8;
then (s +* (I +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) in rng I by A2, FUNCT_1:def 5;
then A4: not (s +* (I +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) destroysdestroy a by A1, SCMFSA7B:def 4;
Y: (ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) /. (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) = (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) by COMPOS_1:38;
T: ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) by AMI_1:123;
thus (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1)) . a = (Following (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) . a by AMI_1:14
.= (Exec ((s +* (I +* (Start-At 0 ,SCM+FSA ))) . (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k))),(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k)) . a by Y, T, AMI_1:54
.= (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) . a by A4, SCMFSA7B:26 ; :: thesis: verum