let s be State of SCM+FSA; :: thesis: for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for k being Nat
for J being really-closed good MacroInstruction of SCM+FSA st ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 & J is_halting_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (times* (a,J)) & ((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ (UsedILoc J))) > 0 holds
((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedILoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedILoc J) \/ FinSeq-Locations)
let p be Instruction-Sequence of SCM+FSA; :: thesis: for a being Int-Location
for k being Nat
for J being really-closed good MacroInstruction of SCM+FSA st ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 & J is_halting_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (times* (a,J)) & ((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ (UsedILoc J))) > 0 holds
((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedILoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedILoc J) \/ FinSeq-Locations)
let a be Int-Location; :: thesis: for k being Nat
for J being really-closed good MacroInstruction of SCM+FSA st ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 & J is_halting_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (times* (a,J)) & ((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ (UsedILoc J))) > 0 holds
((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedILoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedILoc J) \/ FinSeq-Locations)
let k be Nat; :: thesis: for J being really-closed good MacroInstruction of SCM+FSA st ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 & J is_halting_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (times* (a,J)) & ((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ (UsedILoc J))) > 0 holds
((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedILoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedILoc J) \/ FinSeq-Locations)
let J be really-closed good MacroInstruction of SCM+FSA ; :: thesis: ( ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 & J is_halting_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (times* (a,J)) & ((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ (UsedILoc J))) > 0 implies ((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedILoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedILoc J) \/ FinSeq-Locations) )
set UFLI = FinSeq-Locations ;
set I = J;
set ST = StepTimes (a,J,p,s);
set au = 1 -stRWNotIn ({a} \/ (UsedILoc J));
set SW = StepWhile>0 ((1 -stRWNotIn ({a} \/ (UsedILoc J))),(J ";" (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc J))),(intloc 0)))),p,(Exec (((1 -stRWNotIn ({a} \/ (UsedILoc J))) := a),(Initialized s))));
set UILI = UsedILoc J;
assume that
A1: ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 and
A2: J is_halting_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (times* (a,J)) and
A3: ((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ (UsedILoc J))) > 0 ; :: thesis: ((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedILoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedILoc J) \/ FinSeq-Locations)
Macro (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc J))),(intloc 0))) is_halting_on IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k)),p +* (times* (a,J)) by SCMFSA7B:19;
then J ";" (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc J))),(intloc 0))) is_halting_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (times* (a,J)) by A2, SFMASTR1:3;
then A4: DataPart ((StepWhile>0 ((1 -stRWNotIn ({a} \/ (UsedILoc J))),(J ";" (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc J))),(intloc 0)))),p,(Exec (((1 -stRWNotIn ({a} \/ (UsedILoc J))) := a),(Initialized s))))) . (k + 1)) = DataPart (IExec ((J ";" (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc J))),(intloc 0)))),(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) by A1, A3, SCMFSA9A:32;

A5: now :: thesis: for x being Int-Location st x in UsedILoc J holds
((StepTimes (a,J,p,s)) . (k + 1)) . x = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) . x

let x be Int-Location; :: thesis: ( x in UsedILoc J implies ((StepTimes (a,J,p,s)) . (k + 1)) . x = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) . x )
A6: not 1 -stRWNotIn ({a} \/ (UsedILoc J)) in {a} \/ (UsedILoc J) by SCMFSA_M:25;
assume x in UsedILoc J ; :: thesis: ((StepTimes (a,J,p,s)) . (k + 1)) . x = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) . x
then A7: 1 -stRWNotIn ({a} \/ (UsedILoc J)) <> x by A6, XBOOLE_0:def 3;
thus ((StepTimes (a,J,p,s)) . (k + 1)) . x = (IExec ((J ";" (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc J))),(intloc 0)))),(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) . x by A4, SCMFSA_M:2
.= (Exec ((SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc J))),(intloc 0))),(IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))))) . x by A2, SFMASTR1:11
.= (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) . x by A7, SCMFSA_2:65 ; :: thesis: verum

end;

now :: thesis: for x being FinSeq-Location holds ((StepTimes (a,J,p,s)) . (k + 1)) . x = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) . x

let x be FinSeq-Location ; :: thesis: ((StepTimes (a,J,p,s)) . (k + 1)) . x = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) . x
thus ((StepTimes (a,J,p,s)) . (k + 1)) . x = (IExec ((J ";" (SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc J))),(intloc 0)))),(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) . x by A4, SCMFSA_M:2
.= (Exec ((SubFrom ((1 -stRWNotIn ({a} \/ (UsedILoc J))),(intloc 0))),(IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))))) . x by A2, SFMASTR1:12
.= (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) . x by SCMFSA_2:65 ; :: thesis: verum

end;

hence ((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedILoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedILoc J) \/ FinSeq-Locations) by A5, SCMFSA_M:28; :: thesis: verum