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*> = F₁(x,y,z) ) & ( for x, y, z being Element of BOOLEAN holds f2 . <*x,y,z*> = F₁(x,y,z) ) implies f1 = f2 )
assume that
A1: for x, y, z being Element of BOOLEAN holds f1 . <*x,y,z*> = F₁(x,y,z) and
A2: for x, y, z being Element of BOOLEAN holds f2 . <*x,y,z*> = F₁(x,y,z) ; :: thesis: f1 = f2

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

let a be Tuple of 3,BOOLEAN; :: thesis: f1 . a = f2 . a
consider x, y, z being Element of BOOLEAN such that
A3: a = <*x,y,z*> by FINSEQ_2:103;
thus f1 . a = F₁(x,y,z) by A1, A3
.= f2 . a by A2, A3 ; :: thesis: verum

end;

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