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 Th11:
  for s being 0-started State of SCMPDS
  for I being parahalting halt-free Program of SCMPDS st stop I c= P
   holds IC Comput(P, s,LifeSpan(P +* stop I,s)) =  card I
proof
  let s be 0-started State of SCMPDS;
  let I be parahalting halt-free Program of SCMPDS;
  set Css= Comput(P, s,LifeSpan(P,s));
  reconsider n = IC Css as Nat;
  assume
A1: stop I c= P;
  then
A2: P halts_on s by SCMPDS_4:def 7;
A3:  P +* stop I = P by A1,FUNCT_4:98;
  I c= stop I by AFINSQ_1:74;
  then
A4: I c= P by A1;
  now
    assume
A5: IC Css in dom I;
    then I.IC Css=P.IC Css by A4,GRFUNC_1:2
      .=CurInstr(P,Css) by PBOOLE:143
      .=halt SCMPDS by A2,EXTPRO_1:def 15;
    hence contradiction by A5,COMPOS_1:def 27;
  end;
  then
A6: n >= card I by AFINSQ_1:66;
A7: card stop I =card I + 1 by Lm1,AFINSQ_1:17;
  IC Css in dom stop(I) by A1,SCMPDS_4:def 6;
  then n < card I + 1 by A7,AFINSQ_1:66;
  then n <= card I by NAT_1:13;
  hence thesis by A3,A6,XXREAL_0:1;
end;
