reserve x for set,
  D for non empty set,
  k, n for Nat,
  z for Nat;
reserve
  N for with_zero set,
  S for IC-Ins-separated non empty
          with_non-empty_values AMI-Struct over N,
  i for Element of the InstructionsF of S,
  l, l1, l2, l3 for Nat,
  s for State of S;
reserve ss for Element of product the_Values_of S;

theorem Th8:
  for l being Nat st l = z holds SUCC(l,STC N) = {z, z+1}
proof
  let l be Nat such that
A1: l = z;
  set M = STC N;
  set K = the set of all
 NIC(i,l) \ JUMP i where i is Element of the InstructionsF of STC N;
  now
    let y be object;
    hereby
      assume y in K;
      then consider ii being Element of the InstructionsF of STC N such that
A2:   y = NIC(ii,l) \ JUMP ii and
      not contradiction;
      reconsider ii as Instruction of STC N;
      now
        per cases by Th6;
        suppose
A3:       InsCode ii = 1;
          JUMP ii = {} by Lm5;
          then y = {z+1} by A1,A2,A3,Lm4;
          hence y in {{z},{z+1}} by TARSKI:def 2;
        end;
        suppose
A4:       InsCode ii = 0;
          JUMP ii = {} by Lm5;
          then y = {z} by A1,A2,A4,Th2,Th4;
          hence y in {{z},{z+1}} by TARSKI:def 2;
        end;
      end;
      hence y in {{z},{z+1}};
    end;
    assume
A5: y in {{z},{z+1}};
    per cases by A5,TARSKI:def 2;
    suppose
A6:   y = {z};
      halt M = [0,{},{}];
      then reconsider i = [0,0,0] as Instruction of M;
      JUMP i = {} & InsCode i = 0 by Lm5;
      then NIC(i,l) \ JUMP i = y by A1,A6,Th2,Th4;
      hence y in K;
    end;
    suppose
A7:   y = {z+1};
      set i = [1,0,0];
      i in III by TARSKI:def 2;
      then reconsider i as Instruction of M by Def7;
      JUMP i = {} & InsCode i = 1 by Lm5;
      then NIC(i,l) \ JUMP i = y by A1,A7,Lm4;
      hence y in K;
    end;
  end;
  then K = {{z},{z+1}} by TARSKI:2;
  hence thesis by ZFMISC_1:26;
end;
