let f, g be Function of (product (carr G)),REAL ; :: thesis: ( ( for x being Element of product (carr G) holds f . x = |.(normsequence G,x).| ) & ( for x being Element of product (carr G) holds g . x = |.(normsequence G,x).| ) implies f = g )
assume that
A2: for x being Element of product (carr G) holds f . x = |.(normsequence G,x).| and
A3: for x being Element of product (carr G) holds g . x = |.(normsequence G,x).| ; :: thesis: f = g

now

let x be Element of product (carr G); :: thesis: f . x = g . x
f . x = |.(normsequence G,x).| by A2;
hence f . x = g . x by A3; :: thesis: verum

end;

hence f = g by FUNCT_2:113; :: thesis: verum