reserve x for set,
  m,n for Nat,
  a,b,c for Int_position,
  i for Instruction of SCMPDS,
  s,s1,s2 for State of SCMPDS,
  k1,k2 for Integer,
  loc,l1 for Nat,
  I,J for Program of SCMPDS,
  N for with_non-empty_elements set;
reserve P,P1,P2,Q for Instruction-Sequence of SCMPDS;

theorem Th12:
 for s being 0-started State of SCMPDS
  for I being parahalting Program of SCMPDS,k be Nat st
     k < LifeSpan(P +* stop I,s)
 holds IC Comput(P +* stop I,s,k) in dom I
proof let s be 0-started State of SCMPDS;
  let I be parahalting Program of SCMPDS,k be Nat;
  set ss= s, PP = P +* stop I, m=LifeSpan(PP,ss);
  set Sk= Comput(PP, ss,k), Ik=IC Sk;
A1: stop I c= PP by FUNCT_4:25;
  then
A2: PP halts_on ss by SCMPDS_4:def 7;
  reconsider n = Ik as Nat;
A3: Ik in dom stop(I) by A1,SCMPDS_4:def 6;
A4: stop I c= PP by FUNCT_4:25;
  assume
A5: k < m;
A6: now
    assume
A7: n = card I;
A8:  0 in dom Stop SCMPDS by COMPOS_1:3;
A9: (Stop SCMPDS). 0 = halt SCMPDS;
    CurInstr(PP,Sk) =PP.Ik by PBOOLE:143
      .=(stop I).(0+n) by A3,A4,GRFUNC_1:2
      .=halt SCMPDS by A7,A9,A8,AFINSQ_1:def 3;
    hence contradiction by A5,A2,EXTPRO_1:def 15;
  end;
  card stop I=card I + 1 by Lm1,AFINSQ_1:17;
  then n < card I + 1 by A3,AFINSQ_1:66;
  then n <= card I by INT_1:7;
  then n < card I by A6,XXREAL_0:1;
  hence thesis by AFINSQ_1:66;
end;
