reserve R for Ring,
  r for Element of R,
  a, b, d1, d2 for Data-Location of R,
  il, i1, i2 for Nat,
  I for Instruction of SCM R,
  s,s1, s2 for State of SCM R,
  T for InsType of the InstructionsF of SCM R,
  k for Nat;

theorem Th34:
  SUCC(il,SCM R) = {il, il + 1}
proof
  set X = the set of all
 NIC(I, il) \ JUMP I where I is Element of the InstructionsF of SCM
R;
  set N = {il, il + 1};
  now
    let x be object;
    hereby
      assume x in union X;
      then consider Y being set such that
A1:   x in Y and
A2:   Y in X by TARSKI:def 4;
      consider i being Element of the InstructionsF of SCM R such that
A3:   Y = NIC(i, il) \ JUMP i by A2;
      per cases by SCMRING2:7;
      suppose
        i = [0,{},{}];
        then i = halt SCM R;
        then x in {il} \ JUMP halt SCM R by A1,A3,AMISTD_1:2;
        then x = il by TARSKI:def 1;
        hence x in N by TARSKI:def 2;
      end;
      suppose
        ex a,b st i = a:=b;
        then consider a, b such that
A4:     i = a:=b;
        x in {il + 1} \ JUMP (a:=b) by A1,A3,A4,AMISTD_1:12;
        then x = il + 1 by TARSKI:def 1;
        hence x in N by TARSKI:def 2;
      end;
      suppose
        ex a,b st i = AddTo(a,b);
        then consider a, b such that
A5:     i = AddTo(a,b);
        x in {il + 1} \ JUMP AddTo(a,b) by A1,A3,A5,AMISTD_1:12;
        then x = il + 1 by TARSKI:def 1;
        hence x in N by TARSKI:def 2;
      end;
      suppose
        ex a,b st i = SubFrom(a,b);
        then consider a, b such that
A6:     i = SubFrom(a,b);
        x in {il + 1} \ JUMP SubFrom(a,b) by A1,A3,A6,AMISTD_1:12;
        then x = il + 1 by TARSKI:def 1;
        hence x in N by TARSKI:def 2;
      end;
      suppose
        ex a,b st i = MultBy(a,b);
        then consider a, b such that
A7:     i = MultBy(a,b);
        x in {il + 1} \ JUMP MultBy(a,b) by A1,A3,A7,AMISTD_1:12;
        then x = il + 1 by TARSKI:def 1;
        hence x in N by TARSKI:def 2;
      end;
      suppose
        ex i1 st i = goto(i1,R);
        then consider i1 such that
A8:     i = goto(i1,R);
        x in {i1} \ JUMP i by A1,A3,A8,Th29;
        then x in {i1} \ {i1} by A8,Th30;
        hence x in N by XBOOLE_1:37;
      end;
      suppose
        ex a,i1 st i = a=0_goto i1;
        then consider a, i1 such that
A9:     i = a=0_goto i1;
A10:    NIC(i, il) c= {i1, il + 1} by A9,Th31;
        x in NIC(i, il) by A1,A3,XBOOLE_0:def 5;
        then
A11:    x = i1 or x = il + 1 by A10,TARSKI:def 2;
        x in NIC(i, il) \ {i1} by A1,A3,A9,Th33;
        then not x in {i1} by XBOOLE_0:def 5;
        hence x in N by A11,TARSKI:def 1,def 2;
      end;
      suppose
        ex a,r st i = a:=r;
        then consider a, r such that
A12:    i = a := r;
        x in {il + 1} \ JUMP (a:=r) by A1,A3,A12,AMISTD_1:12;
        then x = il + 1 by TARSKI:def 1;
        hence x in N by TARSKI:def 2;
      end;
    end;
    assume
A13: x in {il, il + 1};
    per cases by A13,TARSKI:def 2;
    suppose
A14:  x = il;
      set i = halt SCM R;
      NIC(i, il) \ JUMP i = {il} by AMISTD_1:2;
      then
A15:  {il} in X;
      x in {il} by A14,TARSKI:def 1;
      hence x in union X by A15,TARSKI:def 4;
    end;
    suppose
A16:  x = il + 1;
      set a = the Data-Location of R;
      set i = AddTo(a,a);
      NIC(i, il) \ JUMP i = {il + 1} by AMISTD_1:12;
      then
A17:  {il + 1} in X;
      x in {il + 1} by A16,TARSKI:def 1;
      hence x in union X by A17,TARSKI:def 4;
    end;
  end;
  hence thesis by TARSKI:2;
end;
