reserve m, n for Nat,
  x for set,
  i for Instruction of SCM+FSA,
  I for Program of SCM+FSA,
  a for Int-Location,
  f for FinSeq-Location,
  l, l1 for Nat,
  s,s1,s2 for State of SCM+FSA,
  P,P1,P2 for Instruction-Sequence of SCM+FSA;

theorem Th8:
  for s being 0-started State of SCM+FSA
  for I being parahalting really-closed Program of SCM+FSA
    st I c= P1 & I c= P2
  holds LifeSpan(P1,s) = LifeSpan(P2,s) & Result(P1,s) =  Result(P2,s)
proof
  let s be 0-started State of SCM+FSA;
  let I be parahalting really-closed Program of SCM+FSA;
  assume that
A1: I c= P1 and
A2: I c= P2;
A3: P2 halts_on s by A2,AMISTD_1:def 11;
A4: P1 halts_on s by A1,AMISTD_1:def 11;
A5: now
    let l be Nat;
    assume
A6: CurInstr(P2,Comput(P2,s
,l)) = halt SCM+FSA;
    CurInstr(P1,Comput(P1,s,l))
     = CurInstr(P2,Comput(P2,s,l))
     by Th7,A1,A2;
    hence LifeSpan(P1,s) <= l by A4,A6,EXTPRO_1:def 15;
  end;
  CurInstr(P2,Comput(P2,s,LifeSpan(P1,s)))
   = CurInstr(P1,Comput(P1,s,LifeSpan(P1,s)))
    by Th7,A1,A2
    .= halt SCM+FSA by A4,EXTPRO_1:def 15;
  hence
A7:   LifeSpan(P1,s) = LifeSpan(P2,s) by A5,A3,EXTPRO_1:def 15;
A8: Result(P2,s) = Comput(P2,s,LifeSpan(P1,s))
       by A7,Th1,A2,EXTPRO_1:23;
  Result(P1,s) = Comput(P1,s,LifeSpan(P1,s)) by Th1,A1,EXTPRO_1:23;
  hence thesis by A8,Th7,A1,A2;
end;
