reserve p,q,r for FinSequence,
  x,y for object;

theorem Th21:
  for R being Relation, a,b being object
    holds R reduces a,b iff R[*] reduces a,b
proof
  let R be Relation, a,b be object;
  R reduces a,b iff a = b or [a,b] in R[*] by Th20;
  hence R reduces a,b implies R[*] reduces a,b by Th12,Th15;
  given p being RedSequence of R[*] such that
A1: a = p.1 and
A2: b = p.len p;
  defpred P[Nat] means $1 in dom p implies R reduces a,p.$1;
  now
    let i be Nat such that
A3: i in dom p implies R reduces a,p.i and
A4: i+1 in dom p;
A5: i < len p by A4,Lm2;
    per cases;
    suppose
      i = 0;
      hence R reduces a,p.(i+1) by A1,Th12;
    end;
    suppose
A6:   i > 0;
      then i in dom p by A5,Lm3;
      then [p.i,p.(i+1)] in R[*] by A4,Def2;
      then R reduces p.i, p.(i+1) by Th20;
      hence R reduces a,p.(i+1) by A3,A5,A6,Lm3,Th16;
    end;
  end;
  then
A7: for k being Nat st P[k] holds P[k+1];
A8: len p in dom p by FINSEQ_5:6;
A9: P[ 0 ] by Lm1;
  for i being Nat holds P[i] from NAT_1:sch 2(A9,A7);
  hence thesis by A2,A8;
end;
