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

theorem Th18:
  for R being Relation, a,b being object st R reduces a,b & a <> b
  holds a in field R & b in field R
proof
  let R be Relation, a,b be object;
  given p being RedSequence of R such that
A1: a = p.1 and
A2: b = p.len p;
A3: len p >= 0+1 by NAT_1:13;
  assume a <> b;
  then len p > 1 by A1,A2,A3,XXREAL_0:1;
  then
A4: 1+1 in dom p by Lm4;
  1 in dom p by A3,Lm3;
  then
A5: [a,p.2] in R by A1,A4,Def2;
  hence a in field R by RELAT_1:15;
  defpred P[Nat] means $1 in dom p implies p.$1 in field R;
A6: len p in dom p by FINSEQ_5:6;
  now
    let i be Nat such that
    i in dom p implies p.i in field R and
A7: i+1 in dom p;
A8: i < len p by A7,Lm2;
    per cases;
    suppose
      i = 0;
      hence p.(i+1) in field R by A1,A5,RELAT_1:15;
    end;
    suppose
      i > 0;
      then i in dom p by A8,Lm3;
      then [p.i,p.(i+1)] in R by A7,Def2;
      hence p.(i+1) in field R by RELAT_1:15;
    end;
  end;
  then
A9: for k being Nat st P[k] holds P[k+1];
A10: P[ 0 ] by Lm1;
  for i being Nat holds P[i] from NAT_1:sch 2(A10,A9);
  hence thesis by A2,A6;
end;
