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;

theorem Th5:
  for R being Relation, P being RedSequence of R st len P > 1 holds
  ex Q being RedSequence of R st <*P.1*>^Q = P & len Q + 1 = len P
proof
  let R be Relation, P be RedSequence of R such that
A1: len P > 1;
  consider x being object, Q being FinSequence such that
A2: P = <*x*>^Q and
A3: len P = len Q + 1 by REWRITE1:5;
  1 + len Q > 1 + 0 by A1,A3;
  then len <*x*> = 1 & len Q > 0 by FINSEQ_1:39;
  then
A4: Q is RedSequence of R by A2,Th4;
  P.1 = x by A2,FINSEQ_1:41;
  hence thesis by A2,A3,A4;
end;
