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

theorem
  z in product <%X%> implies ex x st x in X & z = <%x%>
  proof
    assume z in product <%X%>;
    then consider f being Function such that
A1: z = f and
A2: dom f = dom <%X%> and
A3: for x being object st x in dom <%X%> holds f.x in <%X%>.x by CARD_3:def 5;
    reconsider f as XFinSequence by A2,AFINSQ_1:5;
    take f.0;
A4: 0 in {0} by TARSKI:def 1;
A5: <%X%>.0 = X;
    len <%X%> = 1 by AFINSQ_1:34;
    then len f = 1 by A2;
    hence thesis by A5,A4,A1,A2,A3,AFINSQ_1:34,CARD_1:49;
  end;
