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

theorem Th4:
  for p being PartState of SCM+FSA holds intloc 0 in dom Initialized p
proof
  let p be PartState of SCM+FSA;
A1: dom q = {intloc 0} & dom SA0 = {IC SCM+FSA};
  intloc 0 in {intloc 0} by TARSKI:def 1;
  then
A2: intloc 0 in dom p \/ {intloc 0} by XBOOLE_0:def 3;
    Initialized p = Initialize(p +* q) by FUNCT_4:14;
  then dom Initialized p = dom (p +* q) \/ dom SA0 by FUNCT_4:def 1
    .= dom p \/ {intloc 0} \/ {IC SCM+FSA} by A1,FUNCT_4:def 1;
  hence thesis by A2,XBOOLE_0:def 3;
end;
