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