reserve X1,X2,X3,X4 for set;
reserve n for non zero Nat;
reserve X for non-empty n-element FinSequence;

theorem Thm24:
  for X being non-empty 1-element FinSequence, S being SemiringFamily of X
    holds
  SemiringProduct(S)=the set of all product <*s*> where s is Element of S.1
  proof
    let X be non-empty 1-element FinSequence,
    S be SemiringFamily of X;
A1: dom X = {1} by FINSEQ_1:2,FINSEQ_1:89;
A2: S is non-empty
    proof
      assume not S is non-empty;
      then {} in rng S by RELAT_1:def 9;
      then consider a be object such that
A3:   a in dom S and
A4:   S.a = {} by FUNCT_1:def 3;
      a in dom X by Thm16,A3; then
A5:   S.1 = {} by A4,A1,TARSKI:def 1;
      S is SemiringFamily of X & 1 in Seg 1 by FINSEQ_1:3;
      hence contradiction by A5,Def2;
    end;
    then
A6: product S = the set of all <*s*> where s is Element of S.1 by Thm21;
    now
      hereby
        let x be object;
        assume x in SemiringProduct(S);
        then consider f be Function such that
A7:     x = product f and
A8:     f in product S by Def3;
        f in the set of all <*s*> where s is Element of S.1 by A2,A8,Thm21;
        then consider s be Element of S.1 such that
A9:     f = <*s*>;
        thus x in the set of all product <*s*> where
             s is Element of S.1 by A7,A9;
      end;
      let x be object;
      assume x in the set of all product <*s*> where s is Element of S.1;
      then consider s be Element of S.1 such that
A10:  x = product <*s*>;
      set f = <*s*>;
      x = product f & f in product S by A10,A6;
      hence x in SemiringProduct(S) by Def3;
    end;
    then the set of all product <*s*> where
         s is Element of S.1 c= SemiringProduct(S) &
         SemiringProduct(S) c= the set of all product <*s*> where
         s is Element of S.1;
    hence thesis;
  end;
