reserve x for set,
  m,n for Nat,
  a,b for Int_position,
  i,j,k for Instruction of SCMPDS,
  s,s1,s2 for State of SCMPDS,
  k1,k2 for Integer,
  loc,l for Nat,
  I,J,K for Program of SCMPDS;
reserve P,P1,P2,Q for Instruction-Sequence of SCMPDS;

theorem Th13:
for s being 0-started State of SCMPDS
  for I being Program of SCMPDS,i being Nat st
   stop I c= P &
I is_closed_on s,P & I is_halting_on s,P
 & i < LifeSpan(P,s)
  holds IC Comput(P,s,i) in dom I
proof
 let s be 0-started State of SCMPDS;
  let I be Program of SCMPDS,i be Nat;
  set sI = stop I, Ci = Comput(P,s,i), Lc = IC Ci;
  assume that
A1: sI c= P and
A2: I is_closed_on s,P and
A3: I is_halting_on s,P and
A4: i < LifeSpan(P,s);
A5: Start-At(0,SCMPDS) c= s by MEMSTR_0:29;
A6: P +* sI = P by A1,FUNCT_4:98;
A7: s = Initialize s by A5,FUNCT_4:98;
  then
A8: Lc in dom sI by A2,A6,SCMPDS_6:def 2;
A9: P halts_on s by A3,A7,A6,SCMPDS_6:def 3;
  now
    assume
A10: sI.Lc=halt SCMPDS;
    CurInstr(P,Ci) =P.Lc by PBOOLE:143
      .=halt SCMPDS by A8,A1,A10,GRFUNC_1:2;
    hence contradiction by A4,A9,EXTPRO_1:def 15;
  end;
  hence thesis by A8,COMPOS_1:51;
end;
