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
  for I,J being Program of SCM+FSA holds Directed I c= I ";" J
proof
  let I,J be Program of SCM+FSA;
A1: card stop I -' 1 = card I by COMPOS_1:71;
A2: card stop Directed I = card stop I by Lm2;
A3: now
    let x be object;
    assume x in dom Directed I;
    then
A4: x in dom I by FUNCT_4:99;
    dom I misses dom Reloc(J,card I) by COMPOS_1:48;
    then not x in dom Reloc(J,card I) by A4,XBOOLE_0:3;
    hence (Directed I).x = (I ";" J).x by FUNCT_4:11,A1,A2;
  end;
  dom (I ";" J) = dom Directed I \/ dom Reloc(J,card I) by FUNCT_4:def 1,A1,A2;
  then dom Directed I c= dom (I ";" J) by XBOOLE_1:7;
  hence thesis by A3,GRFUNC_1:2;
end;
