reserve a, b for Int_position,
  i for Instruction of SCMPDS,
  l for Element of NAT,
  k, k1, k2 for Integer;

theorem Th18:
  SUCC(l,SCMPDS) = NAT
proof
  thus SUCC(l,SCMPDS) c= NAT;
  let x be object;
  set X = the set of all
 NIC(i,l) \ JUMP i where i is Element of the InstructionsF of SCMPDS;
  assume x in NAT;
  then reconsider x as Element of NAT;
  reconsider xx=x as Element of NAT;
  set i = goto ( xx -  l);
  NIC(i,l) = { |. xx -  l +  l.| } by Th3
    .= {x} by ABSVALUE:def 1;
  then
A1: x in NIC(i,l) \ JUMP i by TARSKI:def 1;
  NIC(i,l) \ JUMP i in X;
  hence thesis by A1,TARSKI:def 4;
end;
