let f1, f2 be Point of (DualSp X); :: thesis: ( ( for x being Point of X holds
( g # x is convergent & lim (g # x) = f1 . x ) ) & ( for x being Point of X holds
( g # x is convergent & lim (g # x) = f2 . x ) ) implies f1 = f2 )
assume B1: ( ( for x being Point of X holds
( g # x is convergent & lim (g # x) = f1 . x ) ) & ( for x being Point of X holds
( g # x is convergent & lim (g # x) = f2 . x ) ) ) ; :: thesis: f1 = f2
B2: ( f1 is Lipschitzian linear-Functional of X & f2 is Lipschitzian linear-Functional of X ) by DUALSP01:def 10;
for x being Point of X holds f1 . x = f2 . x

proof

let x be Point of X; :: thesis: f1 . x = f2 . x
thus f1 . x = lim (g # x) by B1
.= f2 . x by B1 ; :: thesis: verum

end;

hence f1 = f2 by B2, FUNCT_2:def 8; :: thesis: verum