deffunc H_{1}( Tuple of 3,BOOLEAN) -> Element of BOOLEAN = F_{1}(($1 . 1),($1 . 2),($1 . 3));

consider f being Function of (3 -tuples_on BOOLEAN),BOOLEAN such that

A1: for a being Tuple of 3,BOOLEAN holds f . a = H_{1}(a)
from FUNCT_2:sch 4();

_{1}(x,y,z) ) & ( for x, y, z being Element of BOOLEAN holds f2 . <*x,y,z*> = F_{1}(x,y,z) ) implies f1 = f2 )

assume that

A2: for x, y, z being Element of BOOLEAN holds f1 . <*x,y,z*> = F_{1}(x,y,z)
and

A3: for x, y, z being Element of BOOLEAN holds f2 . <*x,y,z*> = F_{1}(x,y,z)
; :: thesis: f1 = f2

consider f being Function of (3 -tuples_on BOOLEAN),BOOLEAN such that

A1: for a being Tuple of 3,BOOLEAN holds f . a = H

hereby :: thesis: for f1, f2 being Function of (3 -tuples_on BOOLEAN),BOOLEAN st ( for x, y, z being Element of BOOLEAN holds f1 . <*x,y,z*> = F_{1}(x,y,z) ) & ( for x, y, z being Element of BOOLEAN holds f2 . <*x,y,z*> = F_{1}(x,y,z) ) holds

f1 = f2

let f1, f2 be Function of (3 -tuples_on BOOLEAN),BOOLEAN; :: thesis: ( ( for x, y, z being Element of BOOLEAN holds f1 . <*x,y,z*> = Ff1 = f2

take f = f; :: thesis: for x, y, z being Element of BOOLEAN holds f . <*x,y,z*> = F_{1}(x,y,z)

let x, y, z be Element of BOOLEAN ; :: thesis: f . <*x,y,z*> = F_{1}(x,y,z)

reconsider a = <*x,y,z*> as Tuple of 3,BOOLEAN by FINSEQ_2:104;

thus f . <*x,y,z*> = F_{1}((a . 1),(a . 2),(a . 3))
by A1

.= F_{1}(x,(a . 2),(a . 3))
by FINSEQ_1:45

.= F_{1}(x,y,(a . 3))
by FINSEQ_1:45

.= F_{1}(x,y,z)
by FINSEQ_1:45
; :: thesis: verum

end;let x, y, z be Element of BOOLEAN ; :: thesis: f . <*x,y,z*> = F

reconsider a = <*x,y,z*> as Tuple of 3,BOOLEAN by FINSEQ_2:104;

thus f . <*x,y,z*> = F

.= F

.= F

.= F

assume that

A2: for x, y, z being Element of BOOLEAN holds f1 . <*x,y,z*> = F

A3: for x, y, z being Element of BOOLEAN holds f2 . <*x,y,z*> = F

now :: thesis: for a being Tuple of 3,BOOLEAN holds f1 . a = f2 . a

hence
f1 = f2
by FUNCT_2:63; :: thesis: verumlet a be Tuple of 3,BOOLEAN; :: thesis: f1 . a = f2 . a

consider x, y, z being Element of BOOLEAN such that

A4: a = <*x,y,z*> by FINSEQ_2:103;

thus f1 . a = F_{1}(x,y,z)
by A2, A4

.= f2 . a by A3, A4 ; :: thesis: verum

end;consider x, y, z being Element of BOOLEAN such that

A4: a = <*x,y,z*> by FINSEQ_2:103;

thus f1 . a = F

.= f2 . a by A3, A4 ; :: thesis: verum