reserve m for Nat;
reserve P,PP,P1,P2 for Instruction-Sequence of SCM+FSA;

theorem
  for s being State of SCM+FSA, I,J being MacroInstruction of SCM+FSA,
      a being
  read-write Int-Location st Directed I is_pseudo-closed_on s,P & Directed J
  is_pseudo-closed_on s,P holds
if=0(a,I,J)
  is_halting_on s,P
proof
  let s be State of SCM+FSA;
  let I,J be MacroInstruction of SCM+FSA;
  let a be read-write Int-Location;
  assume
A1: Directed I is_pseudo-closed_on s,P;
  assume
A2: Directed J is_pseudo-closed_on s,P;
  hereby
    per cases;
    suppose
A3:   s.a = 0;
      thus if=0(a,I,J) is_halting_on s,P by A1,A3,Th28;
    end;
    suppose
A4:   s.a <> 0;
      thus if=0(a,I,J) is_halting_on s,P by A2,A4,Th32;
    end;
  end;
end;
