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 st a <> bb holds
((StepForUp a,bb,cc,I,s) . 0 ) . bb = s . bb
let a be read-write Int-Location ; :: thesis: for bb, cc being Int-Location
for I being Program of SCM+FSA st a <> bb holds
((StepForUp a,bb,cc,I,s) . 0 ) . bb = s . bb
let bb, cc be Int-Location ; :: thesis: for I being Program of SCM+FSA st a <> bb holds
((StepForUp a,bb,cc,I,s) . 0 ) . bb = s . bb
let I be Program of SCM+FSA ; :: thesis: ( a <> bb implies ((StepForUp a,bb,cc,I,s) . 0 ) . bb = 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);
bb in {a,bb,cc} by ENUMSET1:def 1;
then bb in {a,bb,cc} \/ (UsedIntLoc I) by XBOOLE_0:def 3;
then A1: bb <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) by SFMASTR1:21;
assume a <> bb ; :: thesis: ((StepForUp a,bb,cc,I,s) . 0 ) . bb = s . bb
then ((s +* (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1)) +* a,(s . bb)) . bb = (s +* (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1)) . bb by FUNCT_7:34
.= s . bb by A1, FUNCT_7:34 ;
hence ((StepForUp a,bb,cc,I,s) . 0 ) . bb = s . bb by SCMFSA_9:def 5; :: thesis: verum