let s be State of SCM+FSA; :: thesis: for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being really-closed MacroInstruction of SCM+FSA st s . a <= 0 & s . (intloc 0) = 1 holds
(IExec ((times (a,I)),p,s)) | ((UsedILoc I) \/ FinSeq-Locations) = s | ((UsedILoc I) \/ FinSeq-Locations)
let p be Instruction-Sequence of SCM+FSA; :: thesis: for a being Int-Location
for I being really-closed MacroInstruction of SCM+FSA st s . a <= 0 & s . (intloc 0) = 1 holds
(IExec ((times (a,I)),p,s)) | ((UsedILoc I) \/ FinSeq-Locations) = s | ((UsedILoc I) \/ FinSeq-Locations)
let a be Int-Location; :: thesis: for I being really-closed MacroInstruction of SCM+FSA st s . a <= 0 & s . (intloc 0) = 1 holds
(IExec ((times (a,I)),p,s)) | ((UsedILoc I) \/ FinSeq-Locations) = s | ((UsedILoc I) \/ FinSeq-Locations)
let I be really-closed MacroInstruction of SCM+FSA ; :: thesis: ( s . a <= 0 & s . (intloc 0) = 1 implies (IExec ((times (a,I)),p,s)) | ((UsedILoc I) \/ FinSeq-Locations) = s | ((UsedILoc I) \/ FinSeq-Locations) )
set FSL = FinSeq-Locations ;
assume that
A1: s . a <= 0 and
A2: s . (intloc 0) = 1 ; :: thesis: (IExec ((times (a,I)),p,s)) | ((UsedILoc I) \/ FinSeq-Locations) = s | ((UsedILoc I) \/ FinSeq-Locations)
set UILI = UsedILoc I;
set au = 1 -stRWNotIn ({a} \/ (UsedILoc I));
set WH = while>0 ((1 -stRWNotIn ({a} \/ (UsedILoc I))),(I ";" (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc I))),(intloc 0)))));
set s1 = Exec (((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a),(Initialized s));
A3: ( a = intloc 0 or a is read-write ) by SCMFSA_M:def 2;
A4: Exec (((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a),(Initialized s)) = IExec ((Macro ((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a)),p,s) by SCMFSA6C:5;
A5: (Exec (((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a),(Initialized s))) . (1 -stRWNotIn ({a} \/ (UsedILoc I))) = (Initialized s) . a by SCMFSA_2:63
.= s . a by A2, A3, SCMFSA_M:9, SCMFSA_M:37 ;
then A6: while>0 ((1 -stRWNotIn ({a} \/ (UsedILoc I))),(I ";" (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc I))),(intloc 0))))) is_halting_on IExec ((Macro ((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a)),p,s),p by A1, A4, SCMFSA_9:38;
(Exec (((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a),(Initialized s))) . (intloc 0) = (Initialized s) . (intloc 0) by SCMFSA_2:63
.= 1 by SCMFSA_M:9 ;
then A7: DataPart (IExec ((while>0 ((1 -stRWNotIn ({a} \/ (UsedILoc I))),(I ";" (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc I))),(intloc 0)))))),p,(Exec (((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a),(Initialized s))))) = DataPart (Exec (((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a),(Initialized s))) by A1, A5, SCMFSA9A:35;

A8: now :: thesis: for x being FinSeq-Location st x in FinSeq-Locations holds
(IExec ((times (a,I)),p,s)) . x = s . x

let x be FinSeq-Location ; :: thesis: ( x in FinSeq-Locations implies (IExec ((times (a,I)),p,s)) . x = s . x )
assume x in FinSeq-Locations ; :: thesis: (IExec ((times (a,I)),p,s)) . x = s . x
thus (IExec ((times (a,I)),p,s)) . x = (IExec ((while>0 ((1 -stRWNotIn ({a} \/ (UsedILoc I))),(I ";" (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc I))),(intloc 0)))))),p,(Exec (((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a),(Initialized s))))) . x by A4, SFMASTR1:15, A6
.= (Exec (((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a),(Initialized s))) . x by A7, SCMFSA_M:2
.= (Initialized s) . x by SCMFSA_2:63
.= s . x by SCMFSA_M:37 ; :: thesis: verum

end;

A9: DataPart s = DataPart (Initialized s) by A2, SCMFSA_M:19;

A10: now :: thesis: for x being Int-Location st x in UsedILoc I holds
(IExec ((times (a,I)),p,s)) . x = s . x

let x be Int-Location; :: thesis: ( x in UsedILoc I implies (IExec ((times (a,I)),p,s)) . x = s . x )
A11: not 1 -stRWNotIn ({a} \/ (UsedILoc I)) in {a} \/ (UsedILoc I) by SCMFSA_M:25;
assume x in UsedILoc I ; :: thesis: (IExec ((times (a,I)),p,s)) . x = s . x
then A12: 1 -stRWNotIn ({a} \/ (UsedILoc I)) <> x by A11, XBOOLE_0:def 3;
thus (IExec ((times (a,I)),p,s)) . x = (IExec ((while>0 ((1 -stRWNotIn ({a} \/ (UsedILoc I))),(I ";" (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc I))),(intloc 0)))))),p,(Exec (((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a),(Initialized s))))) . x by A4, SFMASTR1:14, A6
.= (Exec (((1 -stRWNotIn ({a} \/ (UsedILoc I))) := a),(Initialized s))) . x by A7, SCMFSA_M:2
.= (Initialized s) . x by A12, SCMFSA_2:63
.= s . x by A9, SCMFSA_M:2 ; :: thesis: verum

end;

[#] FinSeq-Locations = FinSeq-Locations ;
hence (IExec ((times (a,I)),p,s)) | ((UsedILoc I) \/ FinSeq-Locations) = s | ((UsedILoc I) \/ FinSeq-Locations) by A10, A8, SCMFSA_M:27; :: thesis: verum