let s be State of SCM+FSA ; :: thesis: for a being read-write Int-Location
for dd, bb, cc being Int-Location
for I being Program of SCM+FSA st a <> dd & dd in UsedIntLoc I holds
((StepForUp a,bb,cc,I,s) . 0 ) . dd = s . dd
let a be read-write Int-Location ; :: thesis: for dd, bb, cc being Int-Location
for I being Program of SCM+FSA st a <> dd & dd in UsedIntLoc I holds
((StepForUp a,bb,cc,I,s) . 0 ) . dd = s . dd
let dd, bb, cc be Int-Location ; :: thesis: for I being Program of SCM+FSA st a <> dd & dd in UsedIntLoc I holds
((StepForUp a,bb,cc,I,s) . 0 ) . dd = s . dd
let I be Program of SCM+FSA ; :: thesis: ( a <> dd & dd in UsedIntLoc I implies ((StepForUp a,bb,cc,I,s) . 0 ) . dd = s . dd )
assume that
A1: a <> dd and
A2: dd in UsedIntLoc I ; :: thesis: ((StepForUp a,bb,cc,I,s) . 0 ) . dd = s . dd
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
dd in {a,bb,cc} \/ (UsedIntLoc I) by A2, XBOOLE_0:def 3;
then A3: dd <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) by SFMASTR1:21;
set S = (s +* (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1)) +* a,(s . bb);
((s +* (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1)) +* a,(s . bb)) . dd = (s +* (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1)) . dd by A1, FUNCT_7:34
.= s . dd by A3, FUNCT_7:34 ;
hence ((StepForUp a,bb,cc,I,s) . 0 ) . dd = s . dd by SCMFSA_9:def 5; :: thesis: verum