reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem
  <:{}:> = {} & Frege{} = {} .--> {}
proof
A1: dom doms {} = {}" {} by Def1,RELAT_1:38;
  then
A2: product doms {} = {{}} by CARD_3:10,RELAT_1:41;
A3: now
    let x be object;
    assume
A4: x in {{}};
    then
A5: x = {} by TARSKI:def 1;
    then
    ex h st (Frege{}).{} = h & dom h = dom {} & for x st x in
    dom h holds h.x = (uncurry {}).(x,{} .x) by A2,A4,Def6;
    hence (Frege{}).x = {} by A5;
  end;
A6: meet({} qua set) = {} by SETFAM_1:def 1;
  rng doms {} = {} by A1,RELAT_1:38,41;
  then dom <:{}:> = {} by A6,Th25;
  hence <:{}:> = {};
  dom Frege{} = product doms {} by Def6;
  hence thesis by A2,A3,FUNCOP_1:11;
end;
