let s be State of SCM+FSA; :: thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for I being Program of {INT,(INT *)} holds ((StepForUp (a,bb,cc,I,s)) . 0) . a = s . bb
let a be read-write Int-Location ; :: thesis: for bb, cc being Int-Location
for I being Program of {INT,(INT *)} holds ((StepForUp (a,bb,cc,I,s)) . 0) . a = s . bb
let bb, cc be Int-Location ; :: thesis: for I being Program of {INT,(INT *)} holds ((StepForUp (a,bb,cc,I,s)) . 0) . a = s . bb
let I be Program of {INT,(INT *)}; :: thesis: ((StepForUp (a,bb,cc,I,s)) . 0) . a = s . bb
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
set S = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
( (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ';' (AddTo (a,(intloc 0)))) ';' (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . 0 = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)) & a in dom (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) ) by SCMFSA_2:66, SCMFSA_9:def 5;
hence ((StepForUp (a,bb,cc,I,s)) . 0) . a = s . bb by FUNCT_7:33; :: thesis: verum