reserve m for Nat;
reserve P for Instruction-Sequence of SCM+FSA;

theorem
  for i being Instruction of SCM+FSA holds (i = halt SCM+FSA implies (
  Directed Macro i). 0 = goto  2) & (i <> halt SCM+FSA implies (
  Directed Macro i). 0 = i)
proof
A1: dom id the InstructionsF of SCM+FSA = the InstructionsF of SCM+FSA;
  let i be Instruction of SCM+FSA;
A2: (Macro i). 0 = i by COMPOS_1:58;
   0 in { 0, 1} by TARSKI:def 2;
  then
A3:  0 in dom Macro i by COMPOS_1:61;
A4: card Macro i = 2 by COMPOS_1:56;
  hereby
A5: dom id the InstructionsF of SCM+FSA = the InstructionsF of SCM+FSA;
    assume
A6: i = halt SCM+FSA;
A7: i in dom (halt SCM+FSA .--> goto  2) by A6,TARSKI:def 1;
    rng Macro i c= the InstructionsF of SCM+FSA by RELAT_1:def 19;
    hence
    (Directed Macro i). 0 = (((id the InstructionsF of SCM+FSA) +* (
    halt SCM+FSA, goto  2))*Macro i). 0 by A4,FUNCT_7:116
      .= (((id the InstructionsF of SCM+FSA) +* (halt SCM+FSA .--> goto
     2))*Macro i). 0 by A5,FUNCT_7:def 3
      .= ((id the InstructionsF of SCM+FSA) +* (halt SCM+FSA .--> goto
    2)).i by A3,A2,FUNCT_1:13
      .= (halt SCM+FSA .--> goto  2).i by A7,FUNCT_4:13
      .= goto  2 by A6,FUNCOP_1:72;
  end;
  assume i <> halt SCM+FSA;
  then
A9: not i in dom (halt SCM+FSA .--> goto  2) by TARSKI:def 1;
  rng Macro i c= the InstructionsF of SCM+FSA by RELAT_1:def 19;
  hence (Directed Macro i). 0 = (((id the InstructionsF of SCM+FSA) +* (
  halt SCM+FSA, goto  2))* Macro i). 0 by A4,FUNCT_7:116
    .= (((id the InstructionsF of SCM+FSA) +* (halt SCM+FSA .--> goto
  2))* Macro i). 0 by A1,FUNCT_7:def 3
    .= ((id the InstructionsF of SCM+FSA) +* (halt SCM+FSA .--> goto  2
  )).i by A3,A2,FUNCT_1:13
    .= (id the InstructionsF of SCM+FSA).i by A9,FUNCT_4:11
    .= i;
end;
