reserve x for set,
  m,n for Nat,
  a,b,c for Int_position,
  i for Instruction of SCMPDS,
  s,s1,s2 for State of SCMPDS,
  k1,k2 for Integer,
  loc,l1 for Nat,
  I,J for Program of SCMPDS,
  N for with_non-empty_elements set;

theorem Th2:
  dom stop I c= dom stop (I ';' J)
proof
  set sI=stop I, sIJ=stop (I ';'J);
A1: card sIJ=card (I ';' J) +1 by Lm1,AFINSQ_1:17
    .=card I + card J +1 by AFINSQ_1:17
    .=card I + 1 + card J;
  card sI=card I +1 by Lm1,AFINSQ_1:17;
  then
A2: card sI <= card sIJ by A1,NAT_1:11;
    set A = NAT;
    let x be object;
    assume
A3: x in dom sI;
    dom sI c= A by RELAT_1:def 18;
    then reconsider l=x as Nat by A3;
    reconsider n = l as Nat;
    n < card sI by A3,AFINSQ_1:66;
    then n < card sIJ by A2,XXREAL_0:2;
    hence x in dom sIJ by AFINSQ_1:66;
end;
