let s be State of SCM+FSA ; :: thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for f being FinSeq-Location
for I being Program of SCM+FSA holds ((StepForUp a,bb,cc,I,s) . 0 ) . f = s . f
let a be read-write Int-Location ; :: thesis: for bb, cc being Int-Location
for f being FinSeq-Location
for I being Program of SCM+FSA holds ((StepForUp a,bb,cc,I,s) . 0 ) . f = s . f
let bb, cc be Int-Location ; :: thesis: for f being FinSeq-Location
for I being Program of SCM+FSA holds ((StepForUp a,bb,cc,I,s) . 0 ) . f = s . f
let f be FinSeq-Location ; :: thesis: for I being Program of SCM+FSA holds ((StepForUp a,bb,cc,I,s) . 0 ) . f = s . f
let I be Program of SCM+FSA ; :: thesis: ((StepForUp a,bb,cc,I,s) . 0 ) . f = s . f
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);
((s +* (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1)) +* a,(s . bb)) . f = (s +* (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1)) . f by FUNCT_7:34, SCMFSA_2:83
.= s . f by FUNCT_7:34, SCMFSA_2:83 ;
hence ((StepForUp a,bb,cc,I,s) . 0 ) . f = s . f by SCMFSA_9:def 5; :: thesis: verum