reserve F for total
  NAT-defined (the InstructionsF of SCM)-valued Function;

theorem
  for F being total
   NAT-defined (the InstructionsF of SCM)-valued Function
    st <%halt SCM%> c= F
  for s being 0-started State-consisting of <*>INT
   holds F halts_on s & LifeSpan(F,s) = 0 & Result(F,s) = s
proof
  let F be total
   NAT-defined (the InstructionsF of SCM)-valued Function such that
A1: <%halt SCM%> c= F;
  let s be 0-started State-consisting of <*>INT;
  1 = len <%halt SCM%> by AFINSQ_1:34;
  then
  0 in dom<%halt SCM%> by CARD_1:49,TARSKI:def 1;
  then
A2: F.(0+0) = <%halt SCM%>.0 by A1,GRFUNC_1:2
    .= halt SCM;
A3: s = Comput(F,s,0) by EXTPRO_1:2;
 F.IC s = halt SCM by A2,MEMSTR_0:def 11;
  hence
A4: F halts_on s by A3,EXTPRO_1:30;
  dom F = NAT by PARTFUN1:def 2;
  then CurInstr(F,s) = F.IC s by PARTFUN1:def 6
       .= halt SCM by A2,MEMSTR_0:def 11;
  hence LifeSpan(F,s) = 0 by A4,A3,EXTPRO_1:def 15;
  IC s = 0 by MEMSTR_0:def 11;
  hence thesis by A2,A3,EXTPRO_1:7;
end;
