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

theorem Th10:
  for s1,s2 being 0-started State of SCM+FSA,
      I being really-closed Program of SCM+FSA
    st I is_halting_on s1,P1 & I c= P1 & I c= P2 &
       DataPart s1 = DataPart s2
  holds LifeSpan(P1,s1) = LifeSpan(P2,s2)
proof
  let s1,s2 be 0-started State of SCM+FSA;
  let J be really-closed Program of SCM+FSA;
  assume that
A1: J is_halting_on s1,P1 and
A2: J c= P1 and
A3: J c= P2 and
A4: DataPart s1 = DataPart s2;
A5: P1 = P1 +* J by A2,FUNCT_4:98;
  s1 = Initialize s1 by MEMSTR_0:44;
  then
A6: P1 halts_on s1 by A1,A5;
A7: now
    let k be Nat;
    assume
    CurInstr(P2,Comput(P2,s2,k)) = halt SCM+FSA;
    then CurInstr(P1,Comput(P1,s1,k))
     = halt SCM+FSA by A4,Th9,A2,A3;
    hence LifeSpan(P1,s1) <= k by A6,EXTPRO_1:def 15;
  end;
  CurInstr(P1,Comput(P1,s1,LifeSpan(P1,s1)))
   = halt SCM+FSA by A6,EXTPRO_1:def 15;
  then
A8: CurInstr(P2,Comput(P2,s2,LifeSpan(P1,s1)))
 = halt SCM+FSA by A4,Th9,A2,A3;
  then P2 halts_on s2 by EXTPRO_1:29;
  hence thesis by A8,A7,EXTPRO_1:def 15;
end;
