let f1, f2 be Function of (Polynom-Ring A),B; :: thesis: ( ( for p being Polynomial of A holds f1 . p = Ext_eval (p,x) ) & ( for p being Polynomial of A holds f2 . p = Ext_eval (p,x) ) implies f1 = f2 )
assume that
A3: for p being Polynomial of A holds f1 . p = Ext_eval (p,x) and
A4: for p being Polynomial of A holds f2 . p = Ext_eval (p,x) ; :: thesis: f1 = f2

now :: thesis: for y being Element of (Polynom-Ring A) holds f1 . y = f2 . y

let y be Element of (Polynom-Ring A); :: thesis: f1 . y = f2 . y
reconsider p = y as Polynomial of A by POLYNOM3:def 10;
thus f1 . y = Ext_eval (p,x) by A3
.= f2 . y by A4 ; :: thesis: verum

end;

hence f1 = f2 ; :: thesis: verum