reserve m,n for Nat,
  a for Int_position,
  i,j for Instruction of SCMPDS,
  s,s1,s2 for State of SCMPDS,
  k1 for Integer,
  loc for Nat,
  I,J,K for Program of SCMPDS;
reserve P,P1,P2 for Instruction-Sequence of SCMPDS;

theorem
   0 in dom if=0(a,k1,I,J) &  1 in dom if=0(a,k1,I,J)
proof
  set ci=card if=0(a,k1,I,J);
  ci=card I + card J +2 by Lm4;
  then 2 <= ci by NAT_1:12;
  then 1 < ci by XXREAL_0:2;
  hence thesis by AFINSQ_1:66;
end;
