reserve i, I for set,
  f, g, h for Function,
  s for ManySortedSet of I;
reserve G1, G2, G3 for non empty multMagma,
  x1, x2 for Element of G1,
  y1, y2 for Element of G2,
  z1, z2 for Element of G3;

theorem Th28:
  <*x1*> * <*x2*> = <*x1*x2*>
proof
  set G = <*G1*>;
A1: G.1 = G1;
  reconsider l = <*x1*>, p = <*x2*>, lpl = <*x1*> * <*x2*>, lpp = <*x1*x2*> as
  Element of product Carrier G by Def2;
A2: l.1 = x1;
A3: p.1 = x2;
A5: 1 in {1} by TARSKI:def 1;
A6: for k being Nat st 1 <= k & k <= 1 holds lpl.k = lpp.k
  proof
    let k be Nat;
    assume that
A7: 1 <= k and
A8: k <= 1;
    k in Seg 1 by A7,A8;
    then k = 1 by FINSEQ_1:2,TARSKI:def 1;
    hence thesis by A5,A2,A3,A1,Th1;
  end;
  dom lpl = dom Carrier G by CARD_3:9
    .= Seg 1 by FINSEQ_1:2,PARTFUN1:def 2;
  then
A9: len lpl = 1 by FINSEQ_1:def 3;
  len lpp = 1 by FINSEQ_1:39;
  hence thesis by A9,A6;
end;
