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

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