reserve X for set;
reserve x,y,z for set;

theorem Th26:
  for J be RelStr-yielding ManySortedSet of {} holds product J =
  RelStr (#{{}}, id {{}}#)
proof
  let J be RelStr-yielding ManySortedSet of {};
  set IT = product J;
A1: the carrier of IT = product Carrier J by Def4
    .= {{}} by CARD_3:10,PBOOLE:122;
A2: product Carrier J = {{}} by CARD_3:10,PBOOLE:122;
  the InternalRel of product J = id {{}}
  proof
    reconsider x = {}, y = {} as Element of IT by A1,TARSKI:def 1;
    let a,b be object;
    x = {} --> {{}};
    then reconsider f = x, g = y as Function;
    hereby
      assume
A3:   [a,b] in the InternalRel of IT;
      then
A4:   b in the carrier of IT by ZFMISC_1:87;
A5:   a in the carrier of IT by A3,ZFMISC_1:87;
      then a = {} by A1,TARSKI:def 1;
      then a = b by A1,A4,TARSKI:def 1;
      hence [a,b] in id {{}} by A1,A5,RELAT_1:def 10;
    end;
    assume
A6: [a,b] in id {{}};
    then a in {{}} by RELAT_1:def 10;
    then
A7: a = {} by TARSKI:def 1;
    for i be object st i in {} ex R being RelStr, xi,yi being Element of R
    st R = J.i & xi = f.i & yi = g.i & xi <= yi;
    then
A8: x <= y by A1,A2,Def4;
    a = b by A6,RELAT_1:def 10;
    hence thesis by A7,A8,ORDERS_2:def 5;
  end;
  hence thesis by A1;
end;
