theorem Th53:
  for P being RedSequence of ==>.-relation(TS) st P.1 = [x, v] & P
  .len P = [y, w] holds ex u st v = u^w
proof
  let P be RedSequence of ==>.-relation(TS) such that
A1: P.1 = [x, v] and
A2: P.len P = [y, w];
  0 + 1 <= len P by NAT_1:8;
  then 1 in dom P by FINSEQ_3:25;
  then consider u such that
A3: (P.1)`2 = u^w by A2,Th52;
  take u;
  thus thesis by A1,A3;
end;
