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

theorem Th16:
  for R being Relation, a,b,c being object st
    R reduces a,b & R reduces b,c holds R reduces a,c
proof
  let R be Relation, a,b,c be object;
  given p being RedSequence of R such that
A1: p.1 = a and
A2: p.len p = b;
  given q being RedSequence of R such that
A3: q.1 = b and
A4: q.len q = c;
  reconsider r = p$^q as RedSequence of R by A2,A3,Th8;
  take r;
  consider p1 being FinSequence, x being object such that
A5: p = p1^<*x*> by FINSEQ_1:46;
  0+1 <= len q by NAT_1:13;
  then len q in Seg len q by FINSEQ_1:1;
  then
A6: len q in dom q by FINSEQ_1:def 3;
A7: r = p1^q by A5,Th2;
  p1 = {} or len p1 >= 0+1 by NAT_1:13;
  then r = q & p = <*x*> or 1 in Seg len p1 by A5,A7,FINSEQ_1:1,34;
  then 1 in dom p1 or len p = 1 & r = q by FINSEQ_1:40,def 3;
  then r.1 = p1.1 & p1.1 = a or r.1 = b & b = a by A1,A2,A3,A5,A7,
FINSEQ_1:def 7;
  hence r.1 = a;
  len r = len p1+len q by A7,FINSEQ_1:22;
  hence thesis by A4,A7,A6,FINSEQ_1:def 7;
end;
