reserve u,v,x,y,z,X,Y for set;
reserve r,s for Real;

theorem Th6:
  x in X & y in Y implies <%x,y%> in product <%X,Y%>
  proof
    set f = <%X,Y%>;
    set g = <%x,y%>;
    assume
A1: x in X & y in Y;
A2: len f = 2 by AFINSQ_1:38;
A3: len g = dom g;
    now
      let a be object;
      assume a in dom f;
      then a = 0 or a = 1 by A2,TARSKI:def 2,CARD_1:50;
      hence g.a in f.a by A1;
    end;
    hence thesis by A2,A3,CARD_3:9,AFINSQ_1:38;
  end;
