let NORM1, NORM2 be Function of (BoundedLinearFunctionals X),REAL; :: thesis: ( ( for x being object st x in BoundedLinearFunctionals X holds
NORM1 . x = upper_bound (PreNorms (Bound2Lipschitz (x,X))) ) & ( for x being object st x in BoundedLinearFunctionals X holds
NORM2 . x = upper_bound (PreNorms (Bound2Lipschitz (x,X))) ) implies NORM1 = NORM2 )
assume that
A2: for x being object st x in BoundedLinearFunctionals X holds
NORM1 . x = upper_bound (PreNorms (Bound2Lipschitz (x,X))) and
A3: for x being object st x in BoundedLinearFunctionals X holds
NORM2 . x = upper_bound (PreNorms (Bound2Lipschitz (x,X))) ; :: thesis: NORM1 = NORM2
for z being object st z in BoundedLinearFunctionals X holds
NORM1 . z = NORM2 . z

proof

let z be object ; :: thesis: ( z in BoundedLinearFunctionals X implies NORM1 . z = NORM2 . z )
assume A5: z in BoundedLinearFunctionals X ; :: thesis: NORM1 . z = NORM2 . z
NORM1 . z = upper_bound (PreNorms (Bound2Lipschitz (z,X))) by A2, A5;
hence NORM1 . z = NORM2 . z by A3, A5; :: thesis: verum

end;

hence NORM1 = NORM2 ; :: thesis: verum