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

theorem Th7:
  z in product <%X,Y%> implies ex x,y st x in X & y in Y & z = <%x,y%>
  proof
    assume z in product <%X,Y%>;
    then consider f being Function such that
A1: z = f and
A2: dom f = dom <%X,Y%> and
A3: for x being object st x in dom <%X,Y%> holds f.x in <%X,Y%>.x
by CARD_3:def 5;
    reconsider f as XFinSequence by A2,AFINSQ_1:5;
    take f.0, f.1;
A4: 0 in {0,1} & 1 in {0,1} by TARSKI:def 2;
A5: <%X,Y%>.0 = X & <%X,Y%>.1 = Y;
    len <%X,Y%> = 2 by AFINSQ_1:38;
    then len f = 2 by A2;
    hence thesis by A5,A4,A1,A2,A3,AFINSQ_1:38,CARD_1:50;
  end;
