reserve m,n for Nat,
  I for Program of SCM+FSA,
  s,s1,s2 for State of SCM+FSA,
  a for Int-Location,
  f for FinSeq-Location,
  p,p1,p2 for Instruction-Sequence of SCM+FSA;

theorem Th31:
  for s being State of SCM+FSA,
      I,J being really-closed MacroInstruction of SCM+FSA,
      a being read-write Int-Location st s.a <> 0
:::& J is_closed_onInit s,p
   & J  is_halting_onInit s,p
 holds
:::if=0(a,I,J) is_closed_onInit s,p &
if=0(a,I,J)
  is_halting_onInit s,p
proof
  let s be State of SCM+FSA;
  let I,J be really-closed MacroInstruction of SCM+FSA;
  let a be read-write Int-Location;
  set Is = Initialized s;
  assume s.a <> 0;
  then
A1: Is.a <> 0 by SCMFSA_M:37;
  assume J is_halting_onInit s,p;
  then J is_halting_on Is,p by Th27;
  then
  if=0(a,I,J) is_halting_on Is,p by A1,SCMFSA8B:15;
  hence thesis by Th27;
end;
