consider a being Element of A;
set f = (<*> A) .--> a;
A1: dom ((<*> A) .--> a) = {(<*> A)} by FUNCOP_1:19;
A2: dom ((<*> A) .--> a) c= A *

proof

let z be set ; :: according to TARSKI:def 3 :: thesis: ( not z in dom ((<*> A) .--> a) or z in A * )
assume z in dom ((<*> A) .--> a) ; :: thesis: z in A *
then z = <*> A by A1, TARSKI:def 1;
hence z in A * by FINSEQ_1:def 11; :: thesis: verum

end;

A3: rng ((<*> A) .--> a) = {a} by FUNCOP_1:14;
rng ((<*> A) .--> a) c= A

proof

let z be set ; :: according to TARSKI:def 3 :: thesis: ( not z in rng ((<*> A) .--> a) or z in A )
assume z in rng ((<*> A) .--> a) ; :: thesis: z in A
then z = a by A3, TARSKI:def 1;
hence z in A ; :: thesis: verum

end;

then reconsider f = (<*> A) .--> a as PartFunc of (A * ),A by A2, RELSET_1:11;
A4: f is quasi_total

proof

let x, y be FinSequence of A; :: according to UNIALG_1:def 2 :: thesis: ( len x = len y & x in dom f implies y in dom f )
assume that
A5: len x = len y and
A6: x in dom f ; :: thesis: y in dom f
x = <*> A by A1, A6, TARSKI:def 1;
then len x = 0 ;
then y = <*> A by A5;
hence y in dom f by A1, TARSKI:def 1; :: thesis: verum

end;

f is homogeneous

proof

let x, y be FinSequence; :: according to MARGREL1:def 1,UNIALG_1:def 1 :: thesis: ( not x in dom f or not y in dom f or len x = len y )
assume that
A7: x in dom f and
A8: y in dom f ; :: thesis: len x = len y
x = <*> A by A1, A7, TARSKI:def 1;
hence len x = len y by A1, A8, TARSKI:def 1; :: thesis: verum

end;

hence ex b₁ being PartFunc of (A * ),A st
( b₁ is homogeneous & b₁ is quasi_total & not b₁ is empty ) by A4; :: thesis: verum