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

theorem Th3:
  NIC(goto k,l) = { |.k+l.| }
proof
  set s = the State of SCMPDS;
  set i = goto k;
  set t = |.k+l.|;
  set I = i;
  reconsider n = l as Element of NAT;
  hereby
    let x be object;
    assume x in NIC(i,l);
    then consider s being Element of product the_Values_of SCMPDS
    such that
A1: x = IC Exec(i,s) and
A2: IC s = l;
A3: ex m1 being Element of NAT st m1 = IC s & ICplusConst(s,k ) = |.m1+k.|
    by SCMPDS_2:def 18;
    x = t by A1,A2,A3,SCMPDS_2:54;
    hence x in {t} by TARSKI:def 1;
  end;
  let x be object;
   reconsider u = the n-started State of SCMPDS
     as Element of product the_Values_of SCMPDS by CARD_3:107;
A4: IC u = n by MEMSTR_0:def 11;
  consider m1 being Element of NAT such that
A5: m1 = IC u and
A6: ICplusConst(u,k) = |.m1+k.| by SCMPDS_2:def 18;
  assume x in {t};
  then x = |.m1+k.| by A4,A5,TARSKI:def 1
    .= IC Exec(i,u) by A6,SCMPDS_2:54;
  hence thesis by A4;
end;
