let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA holds IExec ((Stop SCM+FSA),P,s) = Initialized s
let s be State of SCM+FSA; :: thesis: IExec ((Stop SCM+FSA),P,s) = Initialized s
set D = Data-Locations ;
set s1 = Initialize (Initialized s);
set P1 = P +* (Stop SCM+FSA);
A1: Stop SCM+FSA c= P +* (Stop SCM+FSA) by FUNCT_4:25;
A2: Initialize (Initialized s) = Comput ((P +* (Stop SCM+FSA)),(Initialize (Initialized s)),0) ;
A3: (P +* (Stop SCM+FSA)) /. (IC (Initialize (Initialized s))) = (P +* (Stop SCM+FSA)) . (IC (Initialize (Initialized s))) by PBOOLE:143;
A4: 0 in dom (Stop SCM+FSA) by COMPOS_1:3;
A5: s +* (Initialize ((intloc 0) .--> 1)) = Initialize (Initialized s) by MEMSTR_0:44;
A6: CurInstr ((P +* (Stop SCM+FSA)),(Initialize (Initialized s))) = (P +* (Stop SCM+FSA)) . 0 by A3, MEMSTR_0:28
.= (Stop SCM+FSA) . 0 by A4, A1, GRFUNC_1:2 ;
then P +* (Stop SCM+FSA) halts_on Initialize (Initialized s) by A2, EXTPRO_1:29;
then A7: IExec ((Stop SCM+FSA),P,s) = Initialize (Initialized s) by A5, A6, A2, EXTPRO_1:def 9;
then A8: DataPart (IExec ((Stop SCM+FSA),P,s)) = DataPart (Initialize (Initialized s))
.= DataPart (Initialized s) by MEMSTR_0:79 ;

hereby :: thesis: verum

A9: now :: thesis: for x being object st x in dom (IExec ((Stop SCM+FSA),P,s)) holds
(IExec ((Stop SCM+FSA),P,s)) . x = (Initialized s) . x

let x be object ; :: thesis: ( x in dom (IExec ((Stop SCM+FSA),P,s)) implies (IExec ((Stop SCM+FSA),P,s)) . b₁ = (Initialized s) . b₁ )
assume A10: x in dom (IExec ((Stop SCM+FSA),P,s)) ; :: thesis: (IExec ((Stop SCM+FSA),P,s)) . b₁ = (Initialized s) . b₁

per cases ( x is Int-Location or x is FinSeq-Location or x = IC ) by A10, SCMFSA_M:1;

suppose A11: x is Int-Location ; :: thesis: (IExec ((Stop SCM+FSA),P,s)) . b₁ = (Initialized s) . b₁

(IExec ((Stop SCM+FSA),P,s)) . x = (Initialized s) . x by A8, A11, SCMFSA_M:2;
hence (IExec ((Stop SCM+FSA),P,s)) . x = (Initialized s) . x ; :: thesis: verum

end;

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

(IExec ((Stop SCM+FSA),P,s)) . x = (Initialized s) . x by A8, A12, SCMFSA_M:2;
hence (IExec ((Stop SCM+FSA),P,s)) . x = (Initialized s) . x ; :: thesis: verum

end;

suppose A13: x = IC ; :: thesis: (IExec ((Stop SCM+FSA),P,s)) . b₁ = (Initialized s) . b₁

then x in {(IC )} by TARSKI:def 1;
then A14: x in dom (Start-At (0,SCM+FSA)) ;
thus (IExec ((Stop SCM+FSA),P,s)) . x = (Initialize (Initialized s)) . (IC ) by A7, A13
.= (Start-At (0,SCM+FSA)) . (IC ) by A13, A14, FUNCT_4:13
.= ((s +* ((intloc 0) .--> 1)) +* (Start-At (0,SCM+FSA))) . x by A13, A14, FUNCT_4:13
.= (s +* (Initialize ((intloc 0) .--> 1))) . x by FUNCT_4:14
.= (Initialized s) . x ; :: thesis: verum

end;

end;

end;

dom (IExec ((Stop SCM+FSA),P,s)) = the carrier of SCM+FSA by PARTFUN1:def 2
.= dom (Initialized s) by PARTFUN1:def 2 ;
hence IExec ((Stop SCM+FSA),P,s) = Initialized s by A9, FUNCT_1:2; :: thesis: verum

end;