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

theorem
  for R being Relation, X being set, a,b being object holds
  R reduces a,b iff R \/ id X reduces a,b
proof
  let R be Relation, X be set, a,b be object;
  thus R reduces a,b implies R \/ id X reduces a,b by Th22,XBOOLE_1:7;
  given p being RedSequence of R \/ id X such that
A1: p.1 = a and
A2: p.len p = b;
  defpred P[Nat] means $1 in dom p implies R reduces a,p.$1;
  now
    let i be Nat;
    assume
A3: i in dom p implies R reduces a,p.i;
    assume
A4: i+1 in dom p;
    per cases;
    suppose
A5:   i in dom p;
      then [p.i, p.(i+1)] in R \/ id X by A4,Def2;
      then [p.i, p.(i+1)] in R or [p.i, p.(i+1)] in id X by XBOOLE_0:def 3;
      then R reduces p.i, p.(i+1) or p.i = p.(i+1) by Th15,RELAT_1:def 10;
      hence R reduces a, p.(i+1) by A3,A5,Th16;
    end;
    suppose
      not i in dom p;
      then i < 0+1 or i > len p & i+1 <= len p by A4,Lm1,Lm3;
      then i = 0 by NAT_1:13;
      hence R reduces a, p.(i+1) by A1,Th12;
    end;
  end;
  then
A6: for k be Nat st P[k] holds P[k+1];
A7: len p in dom p by Lm3;
A8: P[ 0 ] by Lm1;
  for i being Nat holds P[i] from NAT_1:sch 2(A8,A6);
  hence thesis by A2,A7;
end;
