deffunc H₁( Tuple of 2,BOOLEAN ) -> Element of BOOLEAN = F₁(($1 . 1),($1 . 2));
consider f being Function of (2 -tuples_on BOOLEAN ),BOOLEAN such that
A1: for a being Tuple of 2,BOOLEAN holds f . a = H₁(a) from FUNCT_2:sch 4();

hereby :: thesis: for f1, f2 being Function of (2 -tuples_on BOOLEAN ),BOOLEAN st ( for x, y being Element of BOOLEAN holds f1 . <*x,y*> = F₁(x,y) ) & ( for x, y being Element of BOOLEAN holds f2 . <*x,y*> = F₁(x,y) ) holds
f1 = f2

take f = f; :: thesis: for x, y being Element of BOOLEAN holds f . <*x,y*> = F₁(x,y)
let x, y be Element of BOOLEAN ; :: thesis: f . <*x,y*> = F₁(x,y)
reconsider a = <*x,y*> as Tuple of 2,BOOLEAN by FINSEQ_2:121;
thus f . <*x,y*> = F₁((a . 1),(a . 2)) by A1
.= F₁(x,(a . 2)) by FINSEQ_1:61
.= F₁(x,y) by FINSEQ_1:61 ; :: thesis: verum

end;

let f1, f2 be Function of (2 -tuples_on BOOLEAN ),BOOLEAN ; :: thesis: ( ( for x, y being Element of BOOLEAN holds f1 . <*x,y*> = F₁(x,y) ) & ( for x, y being Element of BOOLEAN holds f2 . <*x,y*> = F₁(x,y) ) implies f1 = f2 )
assume that
A2: for x, y being Element of BOOLEAN holds f1 . <*x,y*> = F₁(x,y) and
A3: for x, y being Element of BOOLEAN holds f2 . <*x,y*> = F₁(x,y) ; :: thesis: f1 = f2

now

let a be Tuple of 2,BOOLEAN ; :: thesis: f1 . a = f2 . a
consider x, y being Element of BOOLEAN such that
A4: a = <*x,y*> by FINSEQ_2:120;
thus f1 . a = F₁(x,y) by A2, A4
.= f2 . a by A3, A4 ; :: thesis: verum

end;

hence f1 = f2 by FUNCT_2:113; :: thesis: verum