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
  not (ex x st x in dom f & f.x is Function) implies uncurry f = {} &
  uncurry' f = {}
proof
  assume
A1: not (ex x st x in dom f & f.x is Function);
A2: now
    set t = the Element of dom uncurry f;
    assume dom uncurry f <> {};
    then ex x,g,y st t = [x,y] & x in dom f & g = f.x & y in dom g by Def2;
    hence contradiction by A1;
  end;
  hence uncurry f = {};
  thus thesis by A2,Th1,RELAT_1:41;
end;
