theorem
  [[x, u], [y, v]] in ==>.-relation(TS) implies [[x, u^w], [y, v^w]] in
  ==>.-relation(TS)
proof
  assume [[x, u], [y, v]] in ==>.-relation(TS);
  then x, u ==>. y, v, TS by Def4;
  then x, u^w ==>. y, v^w, TS by Th25;
  hence thesis by Def4;
end;
