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

theorem Lem5:
  for I being non empty set, R being Relation of I
  for r being RedSequence of R
  st len r > 1 holds r.len r in I
  proof
    let I be non empty set;
    let R be Relation of I;
    let r be RedSequence of R;
    assume Z0: len r > 1;
    then consider i being Nat such that
A1: len r = 1+i by NAT_1:10;
    len r >= i >= 1 by Z0,A1,NAT_1:13;
    then i in dom r & i+1 in dom r by A1,FINSEQ_3:25,FINSEQ_5:6;
    then [r.i,r.len r] in R by A1,REWRITE1:def 2;
    hence r.len r in I by ZFMISC_1:87;
  end;
