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 Th52:
  for P being RedSequence of ==>.-relation(TS) st P.len P = [y, w]
  holds for k st k in dom P ex u st (P.k)`2 = u^w
proof
  let P be RedSequence of ==>.-relation(TS);
  assume P.len P = [y, w];
  then
A1:(P.len P)`2 = {}^w .= <%>E^w;
  defpred P[Nat] means len P - $1 in dom P implies ex u st (P.(len P - $1))`2
  = u^w;
A2: now
    let k;
    assume
A3: P[k];
    now
      set len2 = len P - k;
      set len1 = len P - (k + 1);
A4:   len1 + 1 = len2;
      assume
A5:   len P - (k + 1) in dom P;
      thus ex u st (P.(len P - (k + 1)))`2 = u^w
      proof
        per cases;
        suppose
A6:       len P - k in dom P;
          then consider u1 such that
A7:       (P.len2)`2 = u1^w by A3;
A8:       [P.len1, P.len2] in ==>.-relation(TS) by A5,A4,A6,REWRITE1:def 2;
          then consider
          x being Element of TS, v1 being Element of E^omega, y being
          Element of TS, v2 such that
A9:       P.len1 = [x, v1] and
A10:      P.len2 = [y, v2] by Th31;
          x, v1 ==>. y, v2, TS by A8,A9,A10,Def4;
          then consider u2 such that
          x, u2 -->. y, TS and
A11:      v1 = u2^v2;
          take u2^u1;
          (P.len1)`2 = u2^v2 by A9,A11
            .= u2^(u1^w) by A7,A10
            .= u2^u1^w by AFINSQ_1:27;
          hence thesis;
        end;
        suppose
          not len P - k in dom P;
          then len1 = len P - 0 by A5,A4,Th3;
          hence thesis;
        end;
      end;
    end;
    hence P[k + 1];
  end;
A12: P[0] by A1;
A13: for k holds P[k] from NAT_1:sch 2(A12, A2);
  hereby
    let k such that
A14: k in dom P;
    k <= len P by A14,FINSEQ_3:25;
    then consider l such that
A15: k + l = len P by NAT_1:10;
    k + l - l = len P - l by A15;
    hence ex u st (P.k)`2 = u^w by A13,A14;
  end;
end;
