reserve m for Nat;
reserve P,PP,P1,P2 for Instruction-Sequence of SCM+FSA;

theorem Th13:
  for s being State of SCM+FSA,
      I being Program of SCM+FSA
       st I is_pseudo-closed_on s,P
 holds I is_pseudo-closed_on Initialize s,P+*I &
   pseudo-LifeSpan(s,P,I) = pseudo-LifeSpan(Initialize s,P+*I,I)
proof
  let s be State of SCM+FSA;
  let I be Program of SCM+FSA;
  set s2 = Initialize Initialize s,
      P2 = P +* I +* I;
  assume
A1: I is_pseudo-closed_on s,P;
  then
A2: for n being Nat st not IC Comput(P2, s2,n) in dom I holds
  pseudo-LifeSpan(s,P,I) <= n by SCMFSA8A:def 4;
A3: for n being Nat st n < pseudo-LifeSpan(s,P,I)
holds IC
  Comput(P2, s2,n) in dom I by A1,SCMFSA8A:def 4;
  IC Comput(P2, s2,pseudo-LifeSpan(s,P,I)) =  card I by A1,SCMFSA8A:def 4;
  hence
A4: I is_pseudo-closed_on Initialize s,P+*I by A3;
  IC Comput(P2, s2,pseudo-LifeSpan(s,P,I)) =  card I by A1,SCMFSA8A:def 4;
  hence thesis by A2,A4,SCMFSA8A:def 4;
end;
