theorem
  not (ex x,y being object st [x,y] in dom f)
   implies curry f = {} & curry' f = {}
proof
  assume
A1: not ex x,y being object st [x,y] in dom f;
  then proj1 dom f = {} by Th8;
  then dom curry f = {} by Def1;
  hence curry f = {};
  now
    let x,y be object;
    assume [x,y] in dom ~f;
    then [y,x] in dom f by FUNCT_4:42;
    hence contradiction by A1;
  end;
  then proj1 dom ~f = {} by Th8;
  then dom curry ~f = {} by Def1;
  hence thesis;
end;
