let s be State of SCM+FSA; :: thesis: for a being read-write Int-Location
for bb, cc, dd being Int-Location
for I being MacroInstruction of SCM+FSA
for p being Instruction-Sequence of SCM+FSA st a <> dd & dd in UsedILoc I holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd
let a be read-write Int-Location; :: thesis: for bb, cc, dd being Int-Location
for I being MacroInstruction of SCM+FSA
for p being Instruction-Sequence of SCM+FSA st a <> dd & dd in UsedILoc I holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd
let bb, cc, dd be Int-Location; :: thesis: for I being MacroInstruction of SCM+FSA
for p being Instruction-Sequence of SCM+FSA st a <> dd & dd in UsedILoc I holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd
let I be MacroInstruction of SCM+FSA ; :: thesis: for p being Instruction-Sequence of SCM+FSA st a <> dd & dd in UsedILoc I holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd
let p be Instruction-Sequence of SCM+FSA; :: thesis: ( a <> dd & dd in UsedILoc I implies ((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd )
assume that
A1: a <> dd and
A2: dd in UsedILoc I ; :: thesis: ((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedILoc I));
dd in {a,bb,cc} \/ (UsedILoc I) by A2, XBOOLE_0:def 3;
then A3: dd <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedILoc I)) by SCMFSA_M:25;
set S = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedILoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedILoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . dd = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedILoc I))),(((s . cc) - (s . bb)) + 1))) . dd by A1, FUNCT_7:32
.= s . dd by A3, FUNCT_7:32 ;
hence ((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd by SCMFSA_9:def 5; :: thesis: verum