
theorem Th1:
  for G, H being non empty multMagma, x being Element of product <*
  G,H*> ex g being Element of G, h being Element of H st x = <*g,h*>
proof
  let G, H be non empty multMagma, x be Element of product <*G,H*>;
  the carrier of product <*G,H*> = product Carrier <*G,H*> by GROUP_7:def 2;
  then consider g being Function such that
A1: x = g and
A2: dom g = dom Carrier <*G,H*> and
A3: for y being object st y in dom Carrier <*G,H*> holds g.y in Carrier <*G
  ,H*>.y by CARD_3:def 5;
A4: ex R being 1-sorted st R = <*G,H*>.2 & Carrier <*G,H*>.2 = the carrier
  of R by Lm2,PRALG_1:def 15;
A5: dom Carrier <*G,H*> = {1,2} by PARTFUN1:def 2;
  then reconsider g as FinSequence by A2,FINSEQ_1:2,def 2;
  g.2 in Carrier <*G,H*>.2 by A3,A5,Lm2;
  then reconsider h1 = g.2 as Element of H by A4;
A6: ex R being 1-sorted st R = <*G,H*>.1 & Carrier <*G,H*>.1 = the carrier
  of R by Lm1,PRALG_1:def 15;
  g.1 in Carrier <*G,H*>.1 by A3,A5,Lm1;
  then reconsider g1 = g.1 as Element of G by A6;
  take g1, h1;
  len g = 2 by A2,A5,FINSEQ_1:2,def 3;
  hence thesis by A1,FINSEQ_1:44;
end;
