reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;
reserve J for Nat;
reserve n for Nat;
reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem Th121:
  x in product <*X*> iff ex y st y in X & x = <*y*>
proof
A1: dom <*X*> = Seg 1 by FINSEQ_1:def 8;
A2: <*X*>.1 = X;
A3: 1 in Seg 1 by FINSEQ_1:2,TARSKI:def 1;
  thus x in product <*X*> implies ex y st y in X & x = <*y*>
  proof
    assume x in product <*X*>;
    then consider f such that
A4: x = f and
A5: dom f = dom <*X*> and
A6: for x being object st x in dom <*X*> holds f.x in <*X*>.x by CARD_3:def 5;
    reconsider f as FinSequence by A1,A5,FINSEQ_1:def 2;
    take f.1;
    thus thesis by A1,A3,A2,A4,A5,A6,FINSEQ_1:def 8;
  end;
  given y such that
A7: y in X and
A8: x = <*y*>;
A9: now
    let a be object;
    assume a in Seg 1;
    then a = 1 by FINSEQ_1:2,TARSKI:def 1;
    hence <*y*>.a in <*X*>.a by A7;
  end;
  dom <*y*> = Seg 1 by FINSEQ_1:def 8;
  hence thesis by A1,A8,A9,CARD_3:def 5;
end;
