reserve F for Instruction-Sequence of SCM;

theorem
  for F for term being bin-term, aux, n being Element of NAT
   st Shift(SCM-Compile(term, aux)^<%halt SCM%>,n) c= F
  for s being n-started State of SCM
   st aux
  > max_Data-Loc_in term holds F halts_on s &
   (Result(F,s)).dl.aux = term@s &
  LifeSpan(F,s) = len SCM-Compile(term, aux)
proof let F;
  let term be bin-term, aux, n be Element of NAT such that
A1: Shift(SCM-Compile(term, aux)^<%halt SCM%>,n) c= F;
   let s be n-started State of SCM;
  assume
A2: aux > max_Data-Loc_in term;
  Shift(SCM-Compile(term, aux),n) c= F by A1,AFINSQ_1:82;
  then consider i being Element of NAT, u being State of SCM such that
A3: u = Comput(F,s,i+1) and
A4: i+1 = len SCM-Compile(term, aux) and
A5: IC Comput(F,s,i)=(n+i) and
A6: IC u = (n+(i+1)) and
A7: u.dl.aux = term@s and
  for dn being Element of NAT st dn < aux holds s.dl.dn = u.dl.dn
       by A2,Th12;
  len <%halt SCM%> = 1 by AFINSQ_1:34;
  then len (SCM-Compile(term, aux)^<%halt SCM%>) = i+1+1 by A4,AFINSQ_1:17;
  then i+1 < len (SCM-Compile(term, aux)^<%halt SCM%>) by NAT_1:13;
  then i+1 in dom (SCM-Compile(term, aux)^<%halt SCM%>) by AFINSQ_1:86;
   then
A8: F.(n+(i+1))=(SCM-Compile(term, aux)^<%halt SCM%>).(i+1) by A1,VALUED_1:51
    .= halt SCM by A4,AFINSQ_1:36;
  hence F halts_on s by A3,A6,EXTPRO_1:30;
  thus (Result(F,s)).dl.aux = term@s by A3,A6,A7,A8,EXTPRO_1:7;
  (n+i) <> (n+(i+1));
  hence thesis by A3,A4,A5,A6,A8,EXTPRO_1:33;
end;
