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

theorem Thm21:
  for X being non-empty 1-element FinSequence holds
  product X = the set of all <*x*> where x is Element of X.1
  proof
    let X be non-empty 1-element FinSequence;
A1: dom X = {1} by FINSEQ_1:89,FINSEQ_1:2;
A2: X.1 is non empty
    proof
      assume
A3:   X.1 is empty;
      1 in dom X by A1,TARSKI:def 1;
      hence contradiction by A3;
    end;
    len X = 1 by CARD_1:def 7; then
    X = <*X.1*> by FINSEQ_1:40; then
    consider I be Function of X.1, product X such that
    I is one-to-one and
A4: I is onto and
A5: for x be object st x in X.1 holds I.x = <*x*> by A2,PRVECT_3:4;
    now
      hereby
        let t be object;
        assume t in product X;
        then t in rng I by A4,FUNCT_2:def 3;
        then consider a be object such that
A6:     a in dom I and
A7:     t = I.a by FUNCT_1:def 3;
        t = <*a*> by A7,A6,A5;
        hence t in the set of all <*x*> where x is Element of X.1 by A6;
      end;
      let t be object;
      assume t in the set of all <*x*> where x is Element of X.1;
      then consider x be Element of X.1 such that
A9:   t = <*x*>;
      reconsider t1=t as FinSequence by A9;
      dom t1 = Seg 1 by A9,FINSEQ_1:def 8;
      then
A10:  dom t1 = dom X by FINSEQ_1:89;
      for a be object st a in dom X holds t1.a in X.a
      proof
        let a be object;
        assume a in dom X;
        then
A11:    a = 1 by A1,TARSKI:def 1;
        then t1.a is Element of X.1 by A9;
        hence thesis by A2,A11;
      end;
      hence t in product X by A10,CARD_3:def 5;
    end;
    then the set of all <*x*> where x is Element of X.1 c= product X &
      product X c= the set of all <*x*> where x is Element of X.1;
    hence thesis;
  end;
