
theorem Th10:
  for X,Y being non empty multMagma holds
  the carrier of product <*X,Y*>
  = product <* the carrier of X,the carrier of Y *>
  proof
    let X,Y be non empty multMagma;
    set CarrX = the carrier of X;
    set CarrY = the carrier of Y;
    A1: the carrier of product (<*X,Y*>) = product
    Carrier (<*X,Y*>) by GROUP_7:def 2;
    len <*CarrX,CarrY*> = 2 by FINSEQ_1:44; then
    A2: dom <*CarrX,CarrY*> = {1,2} by FINSEQ_1:2,def 3;
    for a be object st a in dom (Carrier (<*X,Y*>))
    holds (Carrier (<*X,Y*>)).a =
    (<* the carrier of X,the carrier of Y *>).a
    proof
      let a be object;
      assume
      A4: a in dom (Carrier (<*X,Y*>));
      per cases by A4,TARSKI:def 2;
      suppose
        A5: a = 1;
        then ex R being 1-sorted st R = (<*X,Y*>).1
        & (Carrier (<*X,Y*>)).1 = the carrier of R
        by A4,PRALG_1:def 15;
        hence thesis by A5;
      end;
      suppose
        A6: a = 2;
        then ex R being 1-sorted st R = (<*X,Y*>).2
        & (Carrier (<*X,Y*>)).2 = the carrier of R by A4,PRALG_1:def 15;
        hence thesis by A6;
      end;
    end;
    hence thesis by PARTFUN1:def 2,A2,FUNCT_1:2,A1;
  end;
