reserve x for set;
reserve k, l for Nat;
reserve p, q for FinSequence;

theorem Th1:
  not k in dom p & k + 1 in dom p implies k = 0
proof
  assume that
A1: not k in dom p and
A2: k + 1 in dom p;
A3: k + 1 <= len p by A2,FINSEQ_3:25;
  per cases by A1,FINSEQ_3:25;
  suppose
    k < 1;
    hence thesis by NAT_1:14;
  end;
  suppose
    k > len p;
    hence thesis by A3,NAT_1:13;
  end;
end;
