theorem Th117:
  for S being ManySortedSign
  for X,Y being ManySortedSet of the carrier of S st X c= Y
  holds X is with_missing_variables implies Y is with_missing_variables
proof
  let S be ManySortedSign;
  let X,Y be ManySortedSet of the carrier of S such that
A1: X c= Y and
A2: X"{{}} c= rng the ResultSort of S;
  let x be object;
  assume
A3: x in Y"{{}};
  then
A4: x in dom Y by FUNCT_1:def 7;
A5: Y.x in {{}} by A3,FUNCT_1:def 7;
A6: dom X = the carrier of S by PARTFUN1:def 2;
A7: Y.x = {} by A5,TARSKI:def 1;
  X.x c= Y.x by A1,A4;
  then X.x = {} by A7;
  then X.x in {{}} by TARSKI:def 1;
  then x in X"{{}} by A4,A6,FUNCT_1:def 7;
  hence thesis by A2;
end;
