let NORM1, NORM2 be Function of the_set_of_l1RealSequences ,REAL ; :: thesis: ( ( for x being set st x in the_set_of_l1RealSequences holds
NORM1 . x = Sum (abs (seq_id x)) ) & ( for x being set st x in the_set_of_l1RealSequences holds
NORM2 . x = Sum (abs (seq_id x)) ) implies NORM1 = NORM2 )
assume that
A1: for x being set st x in the_set_of_l1RealSequences holds
NORM1 . x = Sum (abs (seq_id x)) and
A2: for x being set st x in the_set_of_l1RealSequences holds
NORM2 . x = Sum (abs (seq_id x)) ; :: thesis: NORM1 = NORM2
A3: for z being set st z in the_set_of_l1RealSequences holds
NORM1 . z = NORM2 . z

proof

let z be set ; :: thesis: ( z in the_set_of_l1RealSequences implies NORM1 . z = NORM2 . z )
assume A4: z in the_set_of_l1RealSequences ; :: thesis: NORM1 . z = NORM2 . z
NORM1 . z = Sum (abs (seq_id z)) by A1, A4;
hence NORM1 . z = NORM2 . z by A2, A4; :: thesis: verum

end;

( dom NORM1 = the_set_of_l1RealSequences & dom NORM2 = the_set_of_l1RealSequences ) by FUNCT_2:def 1;
hence NORM1 = NORM2 by A3, FUNCT_1:9; :: thesis: verum