
theorem Th5:
for X be non-empty non empty FinSequence, Y be non empty set holds
 ex K be Function of [:product X,Y:],product(X^<*Y*>)
  st K is bijective
   & for x be FinSequence, y be object
       st x in product X & y in Y holds K.(x,y) = x^<*y*>
proof
   let X be non-empty non empty FinSequence;
   let Y be non empty set;
   consider I be Function of [:product X,Y:],[: product X, product <*Y*>:]
    such that
A1: I is bijective
  & for x be object, y be object
     st x in product X & y in Y holds I.(x,y) = [x,<*y*>] by Th4;
   consider J be Function of [:product X,product <*Y*>:],product (X^<*Y*>)
    such that
A2: J is one-to-one onto
  & for x,y be FinSequence
     st x in product X & y in product <*Y*> holds J.(x,y) = x^y by PRVECT_3:6;
   set K=J*I;
   reconsider K as Function of [:product X,Y:],product (X^<*Y*>);
   take K;
A3:rng J = product (X^<*Y*>) by A2,FUNCT_2:def 3;
   rng I = [:product X,product <*Y*>:] by A1,FUNCT_2:def 3; then
   rng(J*I) = J.:([:product X,product <*Y*>:]) by RELAT_1:127
    .= product (X^<*Y*>) by A3,RELSET_1:22;
   then K is onto by FUNCT_2:def 3;
   hence K is bijective by A1,A2;
   thus for x be FinSequence, y be object st
     x in product X & y in Y holds K.(x,y) = x^<*y*>
   proof
    let x be FinSequence, y be object;
    assume A4:x in product X & y in Y; then
A5: I.(x,y) = [x,<*y*>] by A1;
    [x,y] in [:product X,Y:] by A4,ZFMISC_1:87; then
    I.([x,y]) in [:product X,product <*Y*>:] by FUNCT_2:5; then
    <*y*> in product <*Y*> by A5,ZFMISC_1:87; then
    J.(x,<*y*>) = x^<*y*> by A4,A2;
    hence thesis by A5,A4,ZFMISC_1:87,FUNCT_2:15;
   end;
end;
