reserve m for Nat;
reserve P for Instruction-Sequence of SCM+FSA;

theorem Th10:
  for s being State of SCM+FSA,
  P being Instruction-Sequence of SCM+FSA,
  I being initial Program of
  SCM+FSA st I is_pseudo-closed_on s,P
   holds for n being Nat st n < pseudo-LifeSpan(s,P,I)
    holds IC ( Comput(P+* I,
       Initialize s,n)) in dom I &
  CurInstr(P+*I,Comput(P+*I,
     Initialize s,n)) <>
          halt SCM+FSA
proof
  let s be State of SCM+FSA;
  let P be Instruction-Sequence of SCM+FSA;
  let I be initial Program of SCM+FSA;
  set k = pseudo-LifeSpan(s,P,I);
  assume
A1: I is_pseudo-closed_on s,P;
  then
A2: IC Comput(P+*I, Initialize s,k) = card I by Def3;
  hereby
    let n be Nat;
    assume
A3: n < k;
    hence
A4:  IC Comput(P+*I, Initialize s,n) in dom I by A1,Def3;
    assume
    CurInstr (P+*I,Comput(P+*I, Initialize s,n)) = halt SCM+FSA;
    then
    IC Comput(P+*I, Initialize s,k) = IC Comput(P+*I, Initialize s,n)
       by A3,EXTPRO_1:5;
    hence contradiction by A2,A4;
  end;
end;
