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 a being read-write Int-Location holds (Initialized s).a = s.a) &
  (for f holds (Initialized s).f = s.f)
proof
A1: Initialized s = Initialize(s +* ((intloc 0) .--> 1)) by FUNCT_4:14;
  hereby
    let a be read-write Int-Location;
A3: not a in dom ((intloc 0) .--> 1) by TARSKI:def 1;
    not a in dom SA0 by SCMFSA_2:102;
    hence (Initialized s).a = (s+*((intloc 0) .--> 1)).a by A1,FUNCT_4:11
      .= s.a by A3,FUNCT_4:11;
  end;
  hereby
    let f be FinSeq-Location;
    intloc 0 <> f by SCMFSA_2:58;
    then
A4: not f in dom ((intloc 0) .--> 1) by TARSKI:def 1;
    not f in dom SA0 by SCMFSA_2:103;
    hence (Initialized s).f = (s+*((intloc 0) .--> 1)).f by A1,FUNCT_4:11
      .= s.f by A4,FUNCT_4:11;
  end;
end;
