reserve l, m, n for Nat,
  i,j,k for Instruction of SCM+FSA,
  I,J,K for Program of SCM+FSA;
reserve a,b for Int-Location,
  f for FinSeq-Location,
  s,s1,s2 for State of SCM+FSA;

theorem Th13:
  for I being preProgram of SCM+FSA, k being Element of NAT holds
  Reloc(Directed I,k) = Reloc(I,k) +~ (halt SCM+FSA,goto(card I + k))
proof
  let I be preProgram of SCM+FSA;
  let k be Element of NAT;
A1: dom Reloc(I,k) = {m + k where m is Nat: m in dom I} by COMPOS_1:33;
A2: dom Directed I = dom I by FUNCT_4:99;
A3: dom Reloc(Directed I,k) = {m + k where m is Nat: m in dom I}
        by A2,COMPOS_1:33;
A4: now
    let x be object;
    assume
A5: x in {m + k where m is Nat: m in dom I};
    then consider n being Nat such that
A6: x = n + k and
A7: n in dom I;
    dom Directed I = dom I by FUNCT_4:99;
    then reconsider i = (Directed I). n as Instruction of SCM+FSA by A7,
FUNCT_1:106;
    reconsider i0 = I. n as Instruction of SCM+FSA by A7,FUNCT_1:106;
A8: now
      per cases;
      suppose
A9:    i0 = halt SCM+FSA;
        then
A10:    i = goto  card I by A7,FUNCT_4:106;
A11:    (Reloc(I,k)).x
           = IncAddr(i0,k) by A6,A7,COMPOS_1:35
          .= halt SCM+FSA by A9,COMPOS_0:4;
        then (Reloc(I,k)).x in {halt SCM+FSA} by TARSKI:def 1;
        then (Reloc(I,k)).x in dom (halt SCM+FSA .--> goto
         (card I + k));
        then x in dom ((halt SCM+FSA .--> goto  (card I + k))*
        Reloc(I,k)) by A1,A5,FUNCT_1:11;
        hence  (Reloc(I,k) +~ (halt SCM+FSA,goto(card I + k))).x =
        ((halt SCM+FSA .--> goto  (card I + k))* Reloc(I,k)).x by FUNCT_4:13
          .= (halt SCM+FSA .--> goto  (card I + k)). ((
        Reloc(I,k)).x) by A1,A5,FUNCT_1:13
          .= goto  (card I + k) by A11,FUNCOP_1:72
          .= IncAddr(i,k) by A10,SCMFSA_4:1;
      end;
      suppose
A12:    i0 <> halt SCM+FSA;
A13:    (Reloc(I,k)).x = IncAddr(i0,k) by A6,A7,COMPOS_1:35;
A14:      InsCode halt SCM+FSA = 0 by COMPOS_1:70;
        InsCode i0 <> 0 by A12,SCMFSA_2:95;
        then IncAddr(i0,k) <> halt SCM+FSA by A14,COMPOS_0:def 9;
        then not (Reloc(I,k)).x in {halt SCM+FSA} by A13,TARSKI:def 1;
        then
        not (Reloc(I,k)).x in dom (halt SCM+FSA .--> goto
         (card I + k));
        then not x in dom ((halt SCM+FSA .--> goto  (card I + k)) *
        Reloc(I,k)) by FUNCT_1:11;
        hence (Reloc(I,k) +~ (halt SCM+FSA,goto(card I + k))).x =
        (Reloc(I,k)).x by FUNCT_4:11
          .= IncAddr(i,k) by A12,A13,FUNCT_4:105;
      end;
    end;
    thus (Reloc(Directed I,k)).x
   = (Reloc(I,k) +~ (halt SCM+FSA,goto(card I + k))).x
        by A8,A2,A6,A7,COMPOS_1:35;
  end;
   dom(Reloc(I,k) +~ (halt SCM+FSA,goto(card I + k))) =
{m + k where m is Nat:
  m in dom I} by A1,FUNCT_4:99;
  hence thesis by A3,A4,FUNCT_1:2;
end;
