let s be State of SCM+FSA; :: thesis: 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; :: thesis: 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; :: thesis: 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 ; :: thesis: 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; :: thesis: ( 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 ; :: thesis: ((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; :: thesis: verum