let NORM1, NORM2 be Function of the_set_of_BoundedComplexSequences ,REAL ; :: thesis: ( ( for x being set st x in the_set_of_BoundedComplexSequences holds
NORM1 . x = sup (rng |.(seq_id x).|) ) & ( for x being set st x in the_set_of_BoundedComplexSequences holds
NORM2 . x = sup (rng |.(seq_id x).|) ) implies NORM1 = NORM2 )
assume that
A1: for x being set st x in the_set_of_BoundedComplexSequences holds
NORM1 . x = sup (rng |.(seq_id x).|) and
A2: for x being set st x in the_set_of_BoundedComplexSequences holds
NORM2 . x = sup (rng |.(seq_id x).|) ; :: thesis: NORM1 = NORM2
A3: for z being set st z in the_set_of_BoundedComplexSequences holds
NORM1 . z = NORM2 . z

proof

let z be set ; :: thesis: ( z in the_set_of_BoundedComplexSequences implies NORM1 . z = NORM2 . z )
assume A4: z in the_set_of_BoundedComplexSequences ; :: thesis: NORM1 . z = NORM2 . z
NORM1 . z = sup (rng |.(seq_id z).|) by A1, A4;
hence NORM1 . z = NORM2 . z by A2, A4; :: thesis: verum

end;

( dom NORM1 = the_set_of_BoundedComplexSequences & dom NORM2 = the_set_of_BoundedComplexSequences ) by FUNCT_2:def 1;
hence NORM1 = NORM2 by A3, FUNCT_1:9; :: thesis: verum