let f1, f2 be Function of (2 -tuples_on BOOLEAN),BOOLEAN; :: thesis: ( ( for x, y being Element of BOOLEAN holds f1 . <*x,y*> = F_{1}(x,y) ) & ( for x, y being Element of BOOLEAN holds f2 . <*x,y*> = F_{1}(x,y) ) implies f1 = f2 )

assume that

A1: for x, y being Element of BOOLEAN holds f1 . <*x,y*> = F_{1}(x,y)
and

A2: for x, y being Element of BOOLEAN holds f2 . <*x,y*> = F_{1}(x,y)
; :: thesis: f1 = f2

assume that

A1: for x, y being Element of BOOLEAN holds f1 . <*x,y*> = F

A2: for x, y being Element of BOOLEAN holds f2 . <*x,y*> = F

now :: thesis: for a being Tuple of 2,BOOLEAN holds f1 . a = f2 . a

hence
f1 = f2
by FUNCT_2:63; :: thesis: verumlet a be Tuple of 2,BOOLEAN; :: thesis: f1 . a = f2 . a

consider x, y being Element of BOOLEAN such that

A3: a = <*x,y*> by FINSEQ_2:100;

thus f1 . a = F_{1}(x,y)
by A1, A3

.= f2 . a by A2, A3 ; :: thesis: verum

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

A3: a = <*x,y*> by FINSEQ_2:100;

thus f1 . a = F

.= f2 . a by A2, A3 ; :: thesis: verum