reserve x for set;
reserve k, l for Nat;
reserve p, q for FinSequence;
reserve R for Relation;
reserve p, q for RedSequence of R;

theorem Th3:
  k >= 1 implies p | k is RedSequence of R
proof
  assume
A1: k >= 1;
  per cases;
  suppose
    k >= len p;
    hence thesis by FINSEQ_1:58;
  end;
  suppose
A2: k < len p;
A3: now
A4:   dom (p | k) c= dom p by RELAT_1:60;
      let i be Nat such that
A5:   i in dom (p | k) & i + 1 in dom (p | k);
      (p | k).i = p.i & (p | k).(i + 1) = p.(i + 1) by A5,FUNCT_1:47;
      hence [(p | k).i, (p | k).(i + 1)] in R by A5,A4,REWRITE1:def 2;
    end;
    len (p | k) > 0 by A1,A2,FINSEQ_1:59;
    hence thesis by A3,REWRITE1:def 2;
  end;
end;
