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 Th10:
  for F be associative Group-like multMagma-Family of Seg n,
  x be Element of product F,
  s,t 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 & len t = n
  & (for k be Element of Seg n holds t.k in ProjGroup(F,k))
  & x=Product t holds s=t
  proof
    let F be associative Group-like multMagma-Family of Seg n,
    x be Element of product F, s,t be FinSequence of product F;
    set I = Seg n;
    assume that
    A1: len s = n and
    A2: (for k be Element of I holds s.k in ProjGroup(F,k)) and
    A3: x=Product s and
    A4: len t = n and
    A5: (for k be Element of I holds t.k in ProjGroup(F,k)) and
    A6: x=Product t;
    now let i be Nat;
      assume A7: 1<=i & i<= n; then
      reconsider i0=i as Element of I by FINSEQ_1:1;
      consider si be Element of product F such that
      A8: si=s.i & x.i = si.i by A1,A2,A3,A7,Th9;
      consider ti be Element of product F such that
      A9: ti=t.i & x.i = ti.i by A4,A5,A6,A7,Th9;
      s.i0 in ProjGroup(F,i0) by A2; then
      s.i0 in the carrier of ProjGroup(F,i0) by STRUCT_0:def 5; then
      A10:s.i0 in ProjSet(F,i0) by Def2;
      consider sn be Function,gn be Element of (F.i0) such that
      A11:sn=si & dom sn = I & sn.i0 = gn &
      for k be Element of I st k <> i0 holds sn.k = 1_F.k by A8,A10,Th2;
      t.i0 in ProjGroup(F,i0) by A5; then
      t.i0 in the carrier of ProjGroup(F,i0) by STRUCT_0:def 5;
      then
      A12:t.i0 in ProjSet(F,i0) by Def2;
      consider tn be Function,fn be Element of (F.i0) such that
      A13: tn=ti & dom tn = I & tn.i0 = fn &
      for k be Element of I st k <> i0 holds tn.k = 1_F.k by A9,A12,Th2;
      now let x be object;
        assume x in dom sn; then
        reconsider j=x as Element of I by A11;
        per cases;
        suppose j=i;
          hence sn.x=tn.x by A8,A9,A11,A13;
        end;
        suppose A14: j<>i; then
          sn.j = 1_F.j by A11;
          hence sn.x=tn.x by A14,A13;
        end;
      end;
      hence s.i = t.i by A8,A9,A11,A13,FUNCT_1:2;
    end;
    hence thesis by A1,A4;
  end;
