let f1, f2 be BinOp of {0,1,2}; :: thesis: ( ( for x, y being Element of {0,1,2} holds
( ( x = y implies f1 . (x,y) = x ) & ( x <> y implies f1 . (x,y) = 0 ) ) ) & ( for x, y being Element of {0,1,2} holds
( ( x = y implies f2 . (x,y) = x ) & ( x <> y implies f2 . (x,y) = 0 ) ) ) implies f1 = f2 )
assume that
A1: for x, y being Element of {0,1,2} holds
( ( x = y implies f1 . (x,y) = x ) & ( x <> y implies f1 . (x,y) = 0 ) ) and
A2: for x, y being Element of {0,1,2} holds
( ( x = y implies f2 . (x,y) = x ) & ( x <> y implies f2 . (x,y) = 0 ) ) ; :: thesis: f1 = f2
for x, y being Element of {0,1,2} holds f1 . (x,y) = f2 . (x,y)

proof

let x, y be Element of {0,1,2}; :: thesis: f1 . (x,y) = f2 . (x,y)

per cases ( x = y or x <> y ) ;

suppose A3: x = y ; :: thesis: f1 . (x,y) = f2 . (x,y)

then f1 . (x,y) = x by A1
.= f2 . (x,y) by A2, A3 ;
hence f1 . (x,y) = f2 . (x,y) ; :: thesis: verum

end;

suppose A3: x <> y ; :: thesis: f1 . (x,y) = f2 . (x,y)

then f1 . (x,y) = 0 by A1
.= f2 . (x,y) by A2, A3 ;
hence f1 . (x,y) = f2 . (x,y) ; :: thesis: verum

end;

end;

end;

hence f1 = f2 by BINOP_1:2; :: thesis: verum