reserve X,Y,Z,X1,X2,Y1,Y2 for set, x,y,z,t,x1,x2 for object,
  f,g,h,f1,f2,g1,g2 for Function;

theorem Th8:
  not (ex x,y being object st [x,y] in X)
    implies proj1 X = {} & proj2 X = {}
proof
  set x = the Element of proj2 X;
  assume
A1: not (ex x,y being object st [x,y] in X);
  hereby
    set x = the Element of proj1 X;
    assume proj1 X <> {};
    then ex y being object st [x,y] in X by XTUPLE_0:def 12;
    hence contradiction by A1;
  end;
  assume proj2 X <> {};
  then ex y being object st [y,x] in X by XTUPLE_0:def 13;
  hence thesis by A1;
end;
