reserve a,b for object, I,J for set;

theorem Th13:
  for R being asymmetric transitive Relation of I
  for r being RedSequence of R holds r is one-to-one
  proof
    let R be asymmetric transitive Relation of I;
    let r be RedSequence of R;
    let a,b; assume Z0: a in dom r & b in dom r;
    then reconsider i = a, j = b as Nat;
    assume Z1: r.a = r.b & a <> b;
A1: for i,j being Nat st i > j & i in dom r & j in dom r holds r.i <> r.j
    proof
      let i,j be Nat;
      assume i > j; then
A1:   i >= j+1 by NAT_1:13;
      assume Z3: i in dom r;
      assume Z4: j in dom r;
      defpred P[Nat] means $1 in dom r implies [r.j,r.$1] in R & r.$1 <> r.j;
A2:   P[j+1]
      proof
        assume j+1 in dom r;
        hence [r.j,r.(j+1)] in R by Z4,REWRITE1:def 2;
        hence thesis by PREFER_1:22;
      end;
A3:   for i being Nat st i >= j+1 & P[i] holds P[i+1]
      proof
        let i be Nat; assume
Z5:     i >= j+1 & P[i] & i+1 in dom r;
        then 1 <= j+1 & i < i+1 <= len r by NAT_1:11,13,FINSEQ_3:25;
        then 1 <= i <= len r by Z5,XXREAL_0:2;
        then i in dom r by FINSEQ_3:25;
        then
ZZ:     [r.j,r.i] in R & [r.i,r.(i+1)] in R by Z5,REWRITE1:def 2;
        hence [r.j,r.(i+1)] in R by RELAT_2:31;
        thus thesis by PREFER_1:22,ZZ;
      end;
      for i being Nat st i >= j+1 holds P[i] from NAT_1:sch 8(A2,A3);
      hence r.i <> r.j by A1,Z3;
    end;
    i < j or j < i by Z1,XXREAL_0:1;
    hence thesis by Z0,Z1,A1;
  end;
