let s be State of SCM+FSA; for a being read-write Int-Location
for bb, cc being Int-Location
for I being MacroInstruction of SCM+FSA
for p being Instruction-Sequence of SCM+FSA st a <> cc holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
let a be read-write Int-Location; for bb, cc being Int-Location
for I being MacroInstruction of SCM+FSA
for p being Instruction-Sequence of SCM+FSA st a <> cc holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
let bb, cc be Int-Location; for I being MacroInstruction of SCM+FSA
for p being Instruction-Sequence of SCM+FSA st a <> cc holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
let I be MacroInstruction of SCM+FSA ; for p being Instruction-Sequence of SCM+FSA st a <> cc holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
let p be Instruction-Sequence of SCM+FSA; ( a <> cc implies ((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc )
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedILoc I));
set S = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedILoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
cc in {a,bb,cc}
by ENUMSET1:def 1;
then
cc in {a,bb,cc} \/ (UsedILoc I)
by XBOOLE_0:def 3;
then A1:
cc <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedILoc I))
by SCMFSA_M:25;
assume
a <> cc
; ((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
then ((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedILoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . cc =
(s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedILoc I))),(((s . cc) - (s . bb)) + 1))) . cc
by FUNCT_7:32
.=
s . cc
by A1, FUNCT_7:32
;
hence
((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
by SCMFSA_9:def 5; verum