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 Th4:
  k in dom p implies ex q st len q = k & q.1 = p.1 & q.len q = p.k
proof
  assume
A1: k in dom p;
  set q = p | k;
  take q;
A2: 1 <= k by A1,FINSEQ_3:25;
  hence q is RedSequence of R by Th3;
  k <= len p by A1,FINSEQ_3:25;
  hence len q = k by FINSEQ_1:59;
  hence thesis by A2,FINSEQ_3:112;
end;
