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

theorem Th1:
  (for s being State of SCMPDS st IC s = l
   holds Exec(i,s).IC SCMPDS = IC s + 1) implies NIC(i, l) = {l + 1}
proof
  reconsider I = i as Instruction of SCMPDS;
  reconsider n = l as Element of NAT;
  assume
A1: for s being State of SCMPDS st IC s = l
     holds Exec(i, s).IC SCMPDS = IC s + 1;
  hereby
    let x be object;
    assume x in NIC(i,l);
    then consider s being Element of product the_Values_of SCMPDS
    such that
A2: x = IC Exec(i,s) and
A3: IC s = l;
    x = l+1 by A1,A2,A3;
    hence x in {l+1} by TARSKI:def 1;
  end;
  set t = the l-started State of SCMPDS;
  reconsider t = the l-started State of SCMPDS
     as Element of product the_Values_of SCMPDS by CARD_3:107;
A4: IC t = l by MEMSTR_0:def 11;
  let x be object;
  assume x in {l+1};
  then
A5: x = l+1 by TARSKI:def 1;
   IC Exec(I,t) = l+1 by A1,A4;
  hence thesis by A5,A4;
end;
