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 SCM+FSA 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 SCM+FSA holds ((StepForUp a,bb,cc,I,s) . 0 ) . a = s . bb
let bb, cc be Int-Location ; :: thesis: for I being Program of SCM+FSA holds ((StepForUp a,bb,cc,I,s) . 0 ) . a = s . bb
let I be Program of SCM+FSA ; :: 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);
A1: (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) by SCMFSA_9:def 5;
a in dom (s +* (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1)) by SCMFSA_2:66;
hence ((StepForUp a,bb,cc,I,s) . 0 ) . a = s . bb by A1, FUNCT_7:33; :: thesis: verum