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 Th71:
  ==>.-relation(TS) reduces [x, u], [y, v] implies ==>.-relation(
  TS) reduces [x, u^w], [y, v^w]
proof
  assume ==>.-relation(TS) reduces [x, u], [y, v];
  then consider P being RedSequence of ==>.-relation(TS) such that
A1: P.1 = [x, u] and
A2: P.len P = [y, v] by REWRITE1:def 3;
A3: len P >= 0 + 1 by NAT_1:13;
  then len P in dom P by FINSEQ_3:25;
  then
A4: dim2(P.len P, E) = (P.len P)`2 by A1,Th51
    .= v by A2;
  deffunc F(set) = [(P.$1)`1, dim2(P.$1, E)^w];
  consider Q being FinSequence such that
A5: len Q = len P and
A6: for k st k in dom Q holds Q.k = F(k) from FINSEQ_1:sch 2;
  for k being Nat st k in dom Q & k + 1 in dom Q holds [Q.k, Q.
  (k + 1)] in ==>.-relation(TS)
  proof
    let k be Nat such that
A7: k in dom Q and
A8: k + 1 in dom Q;
    1 <= k + 1 & k + 1 <= len Q by A8,FINSEQ_3:25;
    then
A9: k + 1 in dom P by A5,FINSEQ_3:25;
    1 <= k & k <= len Q by A7,FINSEQ_3:25;
    then
A10: k in dom P by A5,FINSEQ_3:25;
    then [P.k, P.(k + 1)] in ==>.-relation(TS) by A9,REWRITE1:def 2;
    then [[(P.k)`1, (P.k)`2], P.(k + 1)] in ==>.-relation(TS) by A10,A9,Th48;
    then
    [[(P.k)`1, (P.k)`2], [(P.(k + 1))`1, (P.(k + 1))`2]] in ==>.-relation
    (TS) by A10,A9,Th48;
    then [[(P.k)`1, dim2(P.k, E)], [(P.(k + 1))`1, (P.(k + 1))`2]] in
    ==>.-relation(TS) by A1,A10,Th51;
    then [[(P.k)`1, dim2(P.k, E)], [(P.(k + 1))`1, dim2(P.(k + 1), E)]] in
    ==>.-relation(TS) by A1,A9,Th51;
    then (P.k)`1, dim2(P.k, E) ==>. (P.(k + 1))`1, dim2(P.(k + 1), E), TS by
Def4;
    then (P.k)`1, dim2(P.k, E)^w ==>. (P.(k + 1))`1, dim2(P.(k + 1), E)^w, TS
    by Th25;
    then
    [[(P.k)`1, dim2(P.k, E)^w], [(P.(k + 1))`1, dim2(P.(k + 1), E)^w]] in
    ==>.-relation(TS) by Def4;
    then
    [[(P.k)`1, dim2(P.k, E)^w], Q.(k + 1)] in ==>.-relation(TS) by A6,A8;
    hence thesis by A6,A7;
  end;
  then reconsider Q as RedSequence of ==>.-relation(TS) by A5,REWRITE1:def 2;
A11: len Q >= 0 + 1 by NAT_1:13;
  then len Q in dom Q by FINSEQ_3:25;
  then Q.len Q = [(P.len Q)`1, dim2(P.len Q, E)^w] by A6;
  then
A12: Q.len Q = [y, v^w] by A2,A5,A4;
  1 in dom P by A3,FINSEQ_3:25;
  then
A13: dim2(P.1, E) = (P.1)`2 by A1,Th51
    .= u by A1;
  1 in dom Q by A11,FINSEQ_3:25;
  then Q.1 = [(P.1)`1, dim2(P.1, E)^w] by A6
    .= [x, u^w] by A1,A13;
  hence thesis by A12,REWRITE1:def 3;
end;
