reserve x, x1, x2, y, y1, y2, z, z1, z2 for object, X, X1, X2 for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u9, u1, u2, v, v1, v2, w, w1, w2 for Element of E^omega;
reserve F, F1, F2 for Subset of E^omega;
reserve i, k, l, n for Nat;
reserve TS for non empty transition-system over F;
reserve s, s9, s1, s2, t, t1, t2 for Element of TS;
reserve S for Subset of TS;

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;
