reserve a, b, d1, d2 for Data-Location,
  il, i1, i2 for Nat,
  I for Instruction of SCM,
  s, s1, s2 for State of SCM,
  T for InsType of the InstructionsF of SCM,
  k,k1 for Nat;

theorem Th15:
  NIC(SCM-goto k, il) = {k}
proof
  now
    let x be object;
A1: now
      il in NAT by ORDINAL1:def 12;
      then reconsider il1 = il as Element of Values IC SCM by MEMSTR_0:def 6;
      set I = SCM-goto k;
      set t = the State of SCM,
      Q = the Instruction-Sequence of SCM;
      assume
A2:   x = k;
      reconsider n = il as Element of NAT by ORDINAL1:def 12;
      reconsider u = t+*(IC SCM,il1)
       as Element of product the_Values_of SCM by CARD_3:107;
      reconsider P = Q +* (il,I)
       as Instruction-Sequence of SCM;
     reconsider ill=il as Element of NAT by ORDINAL1:def 12;
A3:   P/.ill = P.ill by PBOOLE:143;
    IC SCM in dom t by MEMSTR_0:2;
    then
A4: IC u = n by FUNCT_7:31;
    il in NAT by ORDINAL1:def 12;
    then il in dom Q by PARTFUN1:def 2;
    then
A5: P.n = I by FUNCT_7:31;
    then IC Following(P,u) = k by A3,A4,AMI_3:7;
      hence x in {IC Exec(SCM-goto k,s)
       where s is Element of product the_Values_of SCM
       : IC s = il} by A2,A4,A3,A5;
    end;
    now
      assume x in {IC Exec(SCM-goto k,s)
       where s is Element of product the_Values_of SCM
       : IC s = il};
      then ex s being Element of product the_Values_of SCM
      st x = IC Exec(SCM-goto k,s) & IC s = il;
      hence x = k by AMI_3:7;
    end;
    hence
    x in {k} iff x in {IC Exec(SCM-goto k,s)
       where s is Element of product the_Values_of SCM
     : IC s = il} by A1,TARSKI:def 1;
  end;
  hence thesis by TARSKI:2;
end;
