reserve x for set,
  m,n for Nat,
  a,b for Int_position,
  i,j,k for Instruction of SCMPDS,
  s,s1,s2 for State of SCMPDS,
  k1,k2 for Integer,
  loc,l for Nat,
  I,J,K for Program of SCMPDS;
reserve P,P1,P2,Q for Instruction-Sequence of SCMPDS;

theorem Th39:
  for a be Int_position,i be Integer,n be Nat,I be
  Program of SCMPDS holds card for-down(a,i,n,I)= card I +3
proof
  let a be Int_position,i be Integer,n be Nat, I be Program of
  SCMPDS;
  set i1=(a,i)<=0_goto (card I +3), i2=AddTo(a,i,-n);
  set I4=i1 ';' I, I5=I4 ';' i2;
  card I4=card I+1 by SCMPDS_6:6;
  then card I5=card I +1 +1 by SCMP_GCD:4
    .=card I+ (1+1);
  hence card for-down(a,i,n,I)=card I +2 +1 by SCMP_GCD:4
    .=card I + 3;
end;
