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 Th15:
  for s being 0-started State of SCMPDS
  for I being halt-free Program of SCMPDS st
   stop I c= P &
    I is_halting_on s,P holds LifeSpan(P,s) > 0
proof
  let s be 0-started State of SCMPDS;
  let I be halt-free Program of SCMPDS;
  set si=Initialize s, Pi = P +* stop I;
  assume that
A2: stop I c= P and
A3: I is_halting_on s,P;
A4: Start-At(0,SCMPDS) c= s by MEMSTR_0:29;
A5: P = P +* stop I by A2,FUNCT_4:98;
A6: s=si by A4,FUNCT_4:98;
  assume
  LifeSpan(P,s) <= 0;
  then
A7: LifeSpan(P,s)=0;
A8: I c= stop I by AFINSQ_1:74;
  then
A9: dom I c= dom stop I by RELAT_1:11;
A10:  0 in dom I by AFINSQ_1:66;
A11: Pi/.IC si = Pi.IC si by PBOOLE:143;
A12: stop I c= Pi by FUNCT_4:25;
  Pi halts_on si by A3,SCMPDS_6:def 3;
  then halt SCMPDS
     = CurInstr(Pi,Comput(Pi,si,0)) by A6,A7,A5,EXTPRO_1:def 15
    .= Pi. 0 by A11,MEMSTR_0:def 11
    .= (stop I). 0 by A10,A9,A12,GRFUNC_1:2
    .= I. 0 by A10,A8,GRFUNC_1:2;
  hence contradiction by A10,COMPOS_1:def 27;
end;
