theorem Th51:
  for P being RedSequence of ==>.-relation(TS) st ex x, u st P.1 =
  [x, u] holds for k st k in dom P holds dim2(P.k, E) = (P.k)`2
proof
  let P be RedSequence of ==>.-relation(TS);
  given x, u such that
A1: P.1 = [x, u];
  let k such that
A2: k in dom P;
  per cases;
  suppose
A3: k > 1;
A4: k <= len P by A2,FINSEQ_3:25;
    consider l such that
A5: k = l + 1 by A3,NAT_1:6;
    l <= k by A5,NAT_1:11;
    then
A6: l <= len P by A4,XXREAL_0:2;
    l >= 1 by A3,A5,NAT_1:19;
    then l in dom P by A6,FINSEQ_3:25;
    then [P.l, P.k] in ==>.-relation(TS) by A2,A5,REWRITE1:def 2;
    then
A7: P.k in rng ==>.-relation(TS) by XTUPLE_0:def 13;
    rng ==>.-relation(TS) c= [: the carrier of TS, E^omega :] by RELAT_1:def 19
;
    then
    ex x1, y1 being object
st x1 in the carrier of TS & y1 in E^omega & P .k = [x1, y1]
    by A7,ZFMISC_1:def 2;
    hence thesis by Def5;
  end;
  suppose
A8: k <= 1;
    k >= 1 by A2,FINSEQ_3:25;
    then k = 1 by A8,XXREAL_0:1;
    hence thesis by A1,Def5;
  end;
end;
