deffunc H₁( Element of 1 -tuples_on F₁()) -> Element of F₁() = F₂(($1 . 1));
consider f being Function of (1 -tuples_on F₁()),F₁() such that
A1: for a being Element of 1 -tuples_on F₁() holds f . a = H₁(a) from FUNCT_2:sch 4();

hereby :: thesis: for f1, f2 being Function of (1 -tuples_on F₁()),F₁() st ( for x being Element of F₁() holds f1 . <*x*> = F₂(x) ) & ( for x being Element of F₁() holds f2 . <*x*> = F₂(x) ) holds
f1 = f2

take f = f; :: thesis: for x being Element of F₁() holds f . <*x*> = F₂(x)
let x be Element of F₁(); :: thesis: f . <*x*> = F₂(x)
reconsider a = <*x*> as Element of 1 -tuples_on F₁() by FINSEQ_2:98;
thus f . <*x*> = F₂((a . 1)) by A1
.= F₂(x) ; :: thesis: verum

end;

let f1, f2 be Function of (1 -tuples_on F₁()),F₁(); :: thesis: ( ( for x being Element of F₁() holds f1 . <*x*> = F₂(x) ) & ( for x being Element of F₁() holds f2 . <*x*> = F₂(x) ) implies f1 = f2 )
assume that
A2: for x being Element of F₁() holds f1 . <*x*> = F₂(x) and
A3: for x being Element of F₁() holds f2 . <*x*> = F₂(x) ; :: thesis: f1 = f2

now :: thesis: for a being Element of 1 -tuples_on F₁() holds f1 . a = f2 . a

let a be Element of 1 -tuples_on F₁(); :: thesis: f1 . a = f2 . a
consider x being Element of F₁() such that
A4: a = <*x*> by FINSEQ_2:97;
thus f1 . a = F₂(x) by A2, A4
.= f2 . a by A3, A4 ; :: thesis: verum

end;

hence f1 = f2 by FUNCT_2:63; :: thesis: verum