let NORM1, NORM2 be Function of (ComplexBoundedFunctions (X,Y)),REAL; ( ( for x being object st x in ComplexBoundedFunctions (X,Y) holds
NORM1 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) & ( for x being object st x in ComplexBoundedFunctions (X,Y) holds
NORM2 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) implies NORM1 = NORM2 )
assume that
A2:
for x being object st x in ComplexBoundedFunctions (X,Y) holds
NORM1 . x = upper_bound (PreNorms (modetrans (x,X,Y)))
and
A3:
for x being object st x in ComplexBoundedFunctions (X,Y) holds
NORM2 . x = upper_bound (PreNorms (modetrans (x,X,Y)))
; NORM1 = NORM2
A4:
for z being object st z in ComplexBoundedFunctions (X,Y) holds
NORM1 . z = NORM2 . z
( dom NORM1 = ComplexBoundedFunctions (X,Y) & dom NORM2 = ComplexBoundedFunctions (X,Y) )
by FUNCT_2:def 1;
hence
NORM1 = NORM2
by A4, FUNCT_1:2; verum