let s be State of SCM+FSA ; :: thesis: IExec (Stop SCM+FSA ),s = (Initialized s) +* (Start-At 0 ,SCM+FSA )
set D = Int-Locations \/ FinSeq-Locations ;
set A = NAT ;
set s1 = (Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA ));
A1: (Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA )) = Comput (ProgramPart ((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA ))),0 by AMI_1:13;
A2: IExec (Stop SCM+FSA ),s = (Result (ProgramPart (s +* (Initialized (Stop SCM+FSA )))),(s +* (Initialized (Stop SCM+FSA )))) +* (s | NAT ) by SCMFSA6B:def 1;
Y: (ProgramPart ((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA )))) /. (IC ((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA )))) = ((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA ))) . (IC ((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA )))) by COMPOS_1:38;
y: (Stop SCM+FSA ) . 0 = halt SCM+FSA by AFINSQ_1:38;
x: 0 in dom (Stop SCM+FSA ) by COMPOS_1:45;
A3: s +* (Initialized (Stop SCM+FSA )) = (Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA )) by SCMFSA8A:13;
then A4: CurInstr (ProgramPart ((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA ))) = ((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA ))) . 0 by Y, FUNCT_4:26, SCMFSA6B:34
.= (Stop SCM+FSA ) . 0 by Th26, x ;
then ProgramPart ((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA ))) halts_on (Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA )) by A1, y, AMI_1:146;
then A5: IExec (Stop SCM+FSA ),s = ((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA ))) +* (s | NAT ) by A3, A4, A1, A2, y, AMI_1:def 22;
then A6: DataPart (IExec (Stop SCM+FSA ),s) = DataPart ((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA ))) by Th37
.= DataPart (Initialized s) by SCMFSA8A:11 ;

hereby :: thesis: verum

A7: dom (Start-At 0 ,SCM+FSA ) = {(IC SCM+FSA )} by FUNCOP_1:19;

A8: now

let x be set ; :: thesis: ( x in dom (IExec (Stop SCM+FSA ),s) implies (IExec (Stop SCM+FSA ),s) . b₁ = ((Initialized s) +* (Start-At 0 ,SCM+FSA )) . b₁ )
assume A9: x in dom (IExec (Stop SCM+FSA ),s) ; :: thesis: (IExec (Stop SCM+FSA ),s) . b₁ = ((Initialized s) +* (Start-At 0 ,SCM+FSA )) . b₁

per cases ( x is Int-Location or x is FinSeq-Location or x = IC SCM+FSA or x is Element of NAT ) by A9, SCMFSA6A:35;

suppose A10: x is Int-Location ; :: thesis: (IExec (Stop SCM+FSA ),s) . b₁ = ((Initialized s) +* (Start-At 0 ,SCM+FSA )) . b₁

then x <> IC SCM+FSA by SCMFSA_2:81;
then A11: not x in dom (Start-At 0 ,SCM+FSA ) by A7, TARSKI:def 1;
(IExec (Stop SCM+FSA ),s) . x = (Initialized s) . x by A6, A10, SCMFSA6A:38;
hence (IExec (Stop SCM+FSA ),s) . x = ((Initialized s) +* (Start-At 0 ,SCM+FSA )) . x by A11, FUNCT_4:12; :: thesis: verum

end;

suppose A12: x is FinSeq-Location ; :: thesis: (IExec (Stop SCM+FSA ),s) . b₁ = ((Initialized s) +* (Start-At 0 ,SCM+FSA )) . b₁

then x <> IC SCM+FSA by SCMFSA_2:82;
then A13: not x in dom (Start-At 0 ,SCM+FSA ) by A7, TARSKI:def 1;
(IExec (Stop SCM+FSA ),s) . x = (Initialized s) . x by A6, A12, SCMFSA6A:38;
hence (IExec (Stop SCM+FSA ),s) . x = ((Initialized s) +* (Start-At 0 ,SCM+FSA )) . x by A13, FUNCT_4:12; :: thesis: verum

end;

suppose A14: x = IC SCM+FSA ; :: thesis: (IExec (Stop SCM+FSA ),s) . b₁ = ((Initialized s) +* (Start-At 0 ,SCM+FSA )) . b₁

then x in {(IC SCM+FSA )} by TARSKI:def 1;
then A15: x in dom (Start-At 0 ,SCM+FSA ) by FUNCOP_1:19;
not x in NAT by A14, COMPOS_1:3;
then not x in (dom s) /\ NAT by XBOOLE_0:def 4;
then not x in dom (s | NAT ) by RELAT_1:90;
hence (IExec (Stop SCM+FSA ),s) . x = ((Initialized s) +* ((Stop SCM+FSA ) +* (Start-At 0 ,SCM+FSA ))) . (IC SCM+FSA ) by A5, A14, FUNCT_4:12
.= (((Initialized s) +* (Stop SCM+FSA )) +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA ) by FUNCT_4:15
.= (Start-At 0 ,SCM+FSA ) . (IC SCM+FSA ) by A14, A15, FUNCT_4:14
.= ((Initialized s) +* (Start-At 0 ,SCM+FSA )) . x by A14, A15, FUNCT_4:14 ;
:: thesis: verum

end;

suppose A17: x is Element of NAT ; :: thesis: (IExec (Stop SCM+FSA ),s) . b₁ = ((Initialized s) +* (Start-At 0 ,SCM+FSA )) . b₁

then x <> IC SCM+FSA by COMPOS_1:3;
then A18: not x in dom (Start-At 0 ,SCM+FSA ) by A7, TARSKI:def 1;
(IExec (Stop SCM+FSA ),s) | NAT = s | NAT by A2, PBOOLE:157
.= (Initialized s) | NAT by Th36 ;
then (IExec (Stop SCM+FSA ),s) . x = (Initialized s) . x by A17, SCMFSA6A:36;
hence (IExec (Stop SCM+FSA ),s) . x = ((Initialized s) +* (Start-At 0 ,SCM+FSA )) . x by A18, FUNCT_4:12; :: thesis: verum

end;

end;

end;

dom (IExec (Stop SCM+FSA ),s) = the carrier of SCM+FSA by PARTFUN1:def 4
.= dom ((Initialized s) +* (Start-At 0 ,SCM+FSA )) by PARTFUN1:def 4 ;
hence IExec (Stop SCM+FSA ),s = (Initialized s) +* (Start-At 0 ,SCM+FSA ) by A8, FUNCT_1:9; :: thesis: verum

end;