
theorem Th23:
 for I being really-closed MacroInstruction of SCM+FSA
  holds I ';' goto 0 is really-closed
proof let I be really-closed MacroInstruction of SCM+FSA;
A1: dom Macro goto 0 = { 0,1 } by COMPOS_1:61;
A2: 1 in dom Macro goto 0 & 0 in dom Macro goto 0 by COMPOS_1:60;
  set F = Macro goto 0;
  Macro goto 0 is really-closed
   proof let l be Nat;
    assume
A3:   l in dom F;
     then per cases by A1,TARSKI:def 2;
     suppose
A4:    l = 0;
      then F/.l = F.l by A2,PARTFUN1:def 6
        .= goto 0 by A4,COMPOS_1:58;
     then NIC(F/.l,l) = {0} by SCMFSA10:33;
    hence NIC(F/.l, l) c= dom F by A2,ZFMISC_1:31;
     end;
     suppose
A5:    l = 1;
      then F/.l = F.l by A2,PARTFUN1:def 6
        .= halt SCM+FSA by A5,COMPOS_1:59;
     then NIC(F/.l,l) = {l} by AMISTD_1:2;
    hence NIC(F/.l, l) c= dom F by A3,ZFMISC_1:31;
     end;
   end;
  then I ';' Macro goto 0 is really-closed;
 hence I ';' goto 0 is really-closed;
end;
