theorem
  x in dom S`2 implies v.((@S`2).x) = v.Val_S(v,S).x
proof
  assume x in dom S`2;
  then v.Val_S(v,S).x = Val_S(v,S).x & x in dom @S`2 by Th12,SUBSTUT1:def 2;
  hence thesis by FUNCT_1:13;
end;
