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 Th5:
  for I,J being Program of SCM+FSA, l being Nat
   st l in dom I & I.l <> halt SCM+FSA holds (I ";" J).l = I.l
proof
  let I,J be Program of SCM+FSA, l be Nat such
  that
A1: l in dom I and
A2: I.l <> halt SCM+FSA;
   Reloc(J,card I) = IncAddr(Shift(J,card I),card I) by COMPOS_1:34;
   then
A3: dom Reloc(J, card I) = dom Shift(J,card I) by COMPOS_1:def 21;
A4: card stop I -' 1 = card I by COMPOS_1:71;
A5: card stop Directed I = card stop I by Lm2;
  now
    assume l in dom(Reloc(J, card I));
    then l in { m+card I where m is Nat: m in dom J } by A3,VALUED_1:def 12;
    then consider m being Nat such that
A6: l = m+card I and
    m in dom J;
    m + card I < card I by A1,A6,AFINSQ_1:66;
    hence contradiction by NAT_1:11;
  end;
  hence (I ";" J).l = (Directed I).l by FUNCT_4:11,A4,A5
    .= I.l by A2,FUNCT_4:105;
end;
