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
  for p being PartState of SCM+FSA, a being Int-Location
   st  a <> intloc 0 & not a in dom p
  holds not a in dom Initialized p
proof
  let p be PartState of SCM+FSA;
  let a be Int-Location;
  assume that
A1: a <> intloc 0 and
A2: not a in dom p;
  assume a in dom Initialized p;
  then a in dom p \/ {intloc 0} \/ {IC SCM+FSA} by Th3;
  then
A3: a in (dom p \/ {intloc 0}) or a in {IC SCM+FSA} by XBOOLE_0:def 3;
  per cases by A3,A2,XBOOLE_0:def 3;
  suppose
    a in {intloc 0};
    hence contradiction by A1,TARSKI:def 1;
  end;
  suppose
    a in {IC SCM+FSA};
    then a = IC SCM+FSA by TARSKI:def 1;
    hence contradiction by SCMFSA_2:56;
  end;
end;
