reserve l, m, n for Nat,
  i,j,k for Instruction of SCM+FSA,
  I,J,K for Program of SCM+FSA;

theorem
  Reloc(Directed I, m) = ((id the InstructionsF of
SCM+FSA) +* (halt SCM+FSA .--> goto (m + card I)))* Reloc(I, m)
proof
  rng(halt SCM+FSA .--> goto (card I)) = {goto (card I)} by FUNCOP_1:8;
  then
A1: rng((id the InstructionsF of SCM+FSA) +* (halt SCM+FSA .--> goto (
  card I))) c= rng(id the InstructionsF of SCM+FSA) \/ {goto (card I)} by
FUNCT_4:17;
A2: rng(id the InstructionsF of SCM+FSA) \/ {goto (card I)} = the
  InstructionsF of SCM+FSA by ZFMISC_1:40;
  dom(halt SCM+FSA .--> goto (card I)) = {halt SCM+FSA};
  then
  dom((id the InstructionsF of SCM+FSA) +* (halt SCM+FSA .--> goto (
  card I))) = dom(id the InstructionsF of SCM+FSA) \/ {halt SCM+FSA} by
FUNCT_4:def 1
    .= the InstructionsF of SCM+FSA by ZFMISC_1:40;
  then reconsider
  f = (id the InstructionsF of SCM+FSA) +* (halt SCM+FSA .--> goto
 card I) as Function of the InstructionsF of SCM+FSA, the InstructionsF of
  SCM+FSA by A1,A2,FUNCT_2:def 1,RELSET_1:4;
A3: IncAddr(goto  card I,m) = goto (m + card I) by SCMFSA_4:1;
  dom id the InstructionsF of SCM+FSA = the InstructionsF of SCM+FSA;
  then
A4: f = (id the InstructionsF of SCM+FSA) +* (halt SCM+FSA, goto  card
  I) by FUNCT_7:def 3;
A5: rng I c= the InstructionsF of SCM+FSA by RELAT_1:def 19;
A6:  Reloc(Directed I,m) = IncAddr(Shift(Directed I,m),m)
             by COMPOS_1:34;
A7:  Reloc(I,m) = IncAddr(Shift(I,m),m) by COMPOS_1:34;
  Directed I = f*I by A4,A5,FUNCT_7:116;
  hence Reloc(Directed I, m)
     = IncAddr(f*Shift(I,m),m) by A6,VALUED_1:22
    .= ((id the InstructionsF of SCM+FSA) +* (halt SCM+FSA .--> goto (m
  + card I)))* Reloc(I, m) by A3,A7,COMPOS_1:47;
end;
