reserve a, b, c, a1, a2, b1, b2 for Int-Location,
  l, l1, l2 for Nat,
  f, g, f1, f2 for FinSeq-Location,
  i, j for Instruction of SCM+FSA,
  X, Y for set;
reserve p, r for preProgram of SCM+FSA,
  I, J for Program of SCM+FSA,
  k, m, n for Nat;
reserve L for finite Subset of Int-Locations;

theorem
  a=0_goto l in rng p or a>0_goto l in rng p implies FirstNotUsed p <> a
proof
  assume a=0_goto l in rng p or a>0_goto l in rng p;
  then consider i being Instruction of SCM+FSA such that
A1: i in rng p and
A2: i = a=0_goto l or i = a>0_goto l;
  UsedIntLoc i = {a} by A2,Th16;
  then
A3: {a} c= UsedILoc p by A1,Th19;
  not FirstNotUsed p in UsedILoc p by Th50;
  hence thesis by A3,ZFMISC_1:31;
end;
