
theorem Th73:
  for X being non empty set for R being Relation of X for r being
  RedSequence of R st r.1 in X holds r is FinSequence of X
proof
  let X be non empty set;
  let R be Relation of X;
  let p be RedSequence of R such that
A1: p.1 in X;
  let x be object;
  assume x in rng p;
  then consider i being object such that
A2: i in dom p and
A3: x = p.i by FUNCT_1:def 3;
  reconsider i as Element of NAT by A2;
A4: i >= 1 by A2,FINSEQ_3:25;
  per cases by A4,XXREAL_0:1;
  suppose
    i = 1;
    hence thesis by A1,A3;
  end;
  suppose
    i > 1;
    then i >= 1+1 by NAT_1:13;
    then consider j being Nat such that
A5: i = 1+1+j by NAT_1:10;
A6: i = j+1+1 by A5;
A7: j+1 >= 1 by NAT_1:11;
    i <= len p by A2,FINSEQ_3:25;
    then j+1 < len p by A6,NAT_1:13;
    then j+1 in dom p by A7,FINSEQ_3:25;
    then [p.(j+1), x] in R by A2,A3,A6,REWRITE1:def 2;
    hence thesis by ZFMISC_1:87;
  end;
end;
