reserve
  I for non empty set,
  F for associative Group-like multMagma-Family of I,
  i, j for Element of I;
reserve n for non zero Nat;

theorem Th11:
  for F be associative Group-like multMagma-Family of Seg n,
  x be Element of product F
  ex s be FinSequence of product F st len s = n &
  (for k be Element of Seg n holds s.k in ProjGroup(F,k)) & x=Product s
  proof
    let F be associative Group-like multMagma-Family of Seg n,
    x be Element of product F;
    set I = Seg n;
    defpred P[Nat,set] means
    ex k be Element of I,g be Element of F.k st k=$1 &
    x.k=g & $2=(1_product F)+* (k,g);
    A1: for k be Nat st k in Seg n
    ex z being Element of product F st P[k,z]
    proof
      let k be Nat;
      assume k in Seg n; then
      reconsider k0=k as Element of I;
      A2: the carrier of product F = product Carrier F by GROUP_7:def 2;
      A3: dom Carrier F = I by PARTFUN1:def 2;
      consider Rj being 1-sorted such that
      A4: Rj = F.k0 & (Carrier F).k0 = the carrier of Rj by PRALG_1:def 15;
      reconsider g=x.k0 as Element of F.k0 by A4,A3,A2,CARD_3:9;
      1_product F +* (k0,g) in ProjSet(F,k0) by Def1; then
      reconsider z= 1_product F +* (k0,g) as Element of product F;
      take z;
      thus P[k,z];
    end;
    consider s be FinSequence of product F such that
    A5: dom s= Seg n & for k be Nat st k in Seg n holds P[k,s.k]
    from FINSEQ_1:sch 5(A1);
    take s;
    n in NAT by ORDINAL1:def 12;
    hence A6: len s = n by A5,FINSEQ_1:def 3;
    thus A7: for k be Element of I holds s.k in ProjGroup(F,k)
    proof
      let k be Element of Seg n;
      ex k0 be Element of I,g be Element of F.k0 st k0=k &
      x.k0=g & s.k=(1_product F)+* (k0,g) by A5; then
      A8: s.k in ProjSet(F,k) by Def1;
      the carrier of (ProjGroup(F,k)) = ProjSet(F,k)
      by Def2;
      hence thesis by A8,STRUCT_0:def 5;
    end;
    set y=Product s;
    A9: the carrier of product F = product Carrier F by GROUP_7:def 2;
    A10: dom x = Seg n by A9,PARTFUN1:def 2;
    A11: dom y = Seg n by A9,PARTFUN1:def 2;
    A12: dom (1_product F)= I by A9,PARTFUN1:def 2;
    now let t be object;
      assume t in dom x; then
      A13: t in Seg n by A9,PARTFUN1:def 2; then
      reconsider i=t as Nat;
      1<=i & i<= n by A13,FINSEQ_1:1; then
      A14:  ex si be Element of product F st
      si=s.i & y.i = si.i by Th9,A6,A7;
      ex i1 be Element of I,g be Element of F.i1 st i1=i &
      x.i1=g & s.i=(1_product F)+* (i1,g) by A13,A5;
      hence x.t=y.t by A12,A14,FUNCT_7:31;
    end;
    hence thesis by A10,A11,FUNCT_1:2;
  end;
