reserve m for Nat;
reserve P,PP,P1,P2 for Instruction-Sequence of SCM+FSA;

theorem Th19:
  for a being Int-Location, I,J being MacroInstruction of SCM+FSA, n being
Element of NAT st n < card I + card J + 3 holds n in dom if=0(a,I,J) & if=0(a,I
  ,J).n <> halt SCM+FSA
proof
  let a be Int-Location;
  let I,J be MacroInstruction of SCM+FSA;
  let n be Element of NAT;
  set J1 = a =0_goto  (card J + 3) ";" J ";" Goto  (card I + 1)
  ";" I;
A1: card J1 = card (Macro (a =0_goto  (card J + 3)) ";" J ";" Goto
   (card I + 1)) + card I by SCMFSA6A:21
    .= card (Macro (a =0_goto  (card J + 3)) ";" J) + card Goto
  (card I + 1) + card I by SCMFSA6A:21
    .= card (Macro (a =0_goto  (card J + 3)) ";" J) + 1 + card I by SCMFSA8A:15
    .= card Macro (a =0_goto  (card J + 3)) + card J + 1 + card I by
SCMFSA6A:21
    .= 2 + card J + 1 + card I by COMPOS_1:56
    .= card I + card J + 3;
  assume n < card I + card J + 3;
  then  n in dom J1 by A1,AFINSQ_1:66;
  then
A2:  n in dom Directed J1 by FUNCT_4:99;
  then
A3: (Directed J1). n in rng Directed J1 by FUNCT_1:def 3;
A4: Directed J1 c= if=0(a,I,J) by SCMFSA6A:16;
  then dom Directed J1 c= dom if=0(a,I,J) by GRFUNC_1:2;
  hence n in dom if=0(a,I,J) by A2;
  if=0(a,I,J). n = (Directed J1). n by A2,A4,GRFUNC_1:2;
  hence thesis by A3,COMPOS_1:def 11;
end;
