reserve m, n for Nat,
  x for set,
  i for Instruction of SCM+FSA,
  I for Program of SCM+FSA,
  a for Int-Location,
  f for FinSeq-Location,
  l, l1 for Nat,
  s,s1,s2 for State of SCM+FSA,
  P,P1,P2 for Instruction-Sequence of SCM+FSA;

theorem
  for I being keeping_0 parahalting Program of SCM+FSA
   holds IExec(I,P,s).intloc 0 = 1
proof
  let I be keeping_0 parahalting Program of SCM+FSA;
A1: Initialize((intloc 0).-->1) c= s+*Initialize((intloc 0).-->1)
 by FUNCT_4:25;
A2: I c= P+*I by FUNCT_4:25;
  P+*I halts_on s +* Initialize((intloc 0).-->1)
   by Th2,A2,A1;
  then
A3:
  ex n st Result(P+*I,s +* Initialize((intloc 0).-->1))
   = Comput(P+*I,s +* Initialize((intloc 0).-->1),n) &
   CurInstr(P+*I,Result(P+*I,s +* Initialize((intloc 0).-->1)))
    = halt SCM+FSA by EXTPRO_1:def 9;
  thus IExec(I,P,s).intloc 0
     = (Initialized s).intloc 0 by A3,Def2,A2
    .= 1 by SCMFSA_M:9;
end;
