reserve x for set,
  D for non empty set,
  k, n for Element of 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 Element of NAT,
  s for State of S;
reserve ss for Element of product the_Values_of S;
reserve T for weakly_standard
 IC-Ins-separated non empty
  with_non-empty_values AMI-Struct over N;

theorem
  for l being Element of NAT holds SUCC(l,STC N) = {l, NextLoc(l,STC N)}
proof
  let l be Element of NAT;
  set M = STC N;
  consider k being Nat such that
A1: l = il.(M,k) by Th6;
A2: k = locnum(l,M) by A1,Def5;
  thus SUCC(l,STC N) = {k,k+1} by A1,Th17,AMISTD_1:8
    .= {k,il.(M,k+1)} by Th17
    .= {l, NextLoc(l,STC N)} by A1,A2,Th17;
end;
