reserve l, m, n for Nat;
reserve a,b for Int-Location,
  f for FinSeq-Location,
  s,s1,s2 for State of SCM+FSA;
reserve L for finite Subset of Int-Locations;
reserve L for finite Subset of FinSeq-Locations;
reserve L for finite Subset of Int-Locations;

theorem
  for w being FinSequence of INT,f be FinSeq-Location,s be State of SCM+FSA
   st  Initialized(f .--> w) c= s
  holds s.f = w & s.(intloc 0) = 1
proof
  let w be FinSequence of INT,f be FinSeq-Location,
  s be State of SCM+FSA;
  set t= f.--> w, p=Initialized t;
  assume
A1:  p c= s;
   reconsider pt = p  as PartState of SCM+FSA;
A2: f in dom t by TARSKI:def 1;
A3: f in dom  pt by A2,FUNCT_4:12;
A4: intloc 0 in dom  pt by Th4;
  ex i being Nat st f = fsloc i by SCMFSA_2:9;
    then f <> intloc 0 by SCMFSA_2:99;
    then not f in {intloc 0} by TARSKI:def 1;
    then
A5: not f in dom((intloc 0) .--> 1);
A6: dom Initialize((intloc 0) .--> 1)
     = dom ((intloc 0) .--> 1) \/ dom Start-At(0,SCM+FSA) by FUNCT_4:def 1;
    not f in dom Start-At(0,SCM+FSA) by SCMFSA_2:103;
    then
A7: not f in dom Initialize ((intloc 0) .--> 1) by A5,A6,XBOOLE_0:def 3;
  thus s.f = pt.f by A1,A3,GRFUNC_1:2
    .= t.f by A7,FUNCT_4:11
    .= w by FUNCOP_1:72;
  thus s.intloc 0 =p.intloc 0 by A1,A4,GRFUNC_1:2
    .= 1 by Th12,Th10,FUNCT_4:13;
end;
