let NORM1, NORM2 be Function of the_set_of_l1RealSequences,REAL; :: thesis: ( ( for x being object st x in the_set_of_l1RealSequences holds
NORM1 . x = Sum (abs (seq_id x)) ) & ( for x being object st x in the_set_of_l1RealSequences holds
NORM2 . x = Sum (abs (seq_id x)) ) implies NORM1 = NORM2 )
assume that
A2: for x being object st x in the_set_of_l1RealSequences holds
NORM1 . x = Sum (abs (seq_id x)) and
A3: for x being object st x in the_set_of_l1RealSequences holds
NORM2 . x = Sum (abs (seq_id x)) ; :: thesis: NORM1 = NORM2
A4: for z being object st z in the_set_of_l1RealSequences holds
NORM1 . z = NORM2 . z

proof

let z be object ; :: thesis: ( z in the_set_of_l1RealSequences implies NORM1 . z = NORM2 . z )
assume A5: z in the_set_of_l1RealSequences ; :: thesis: NORM1 . z = NORM2 . z
NORM1 . z = Sum (abs (seq_id z)) by A2, A5;
hence NORM1 . z = NORM2 . z by A3, A5; :: thesis: verum

end;

( dom NORM1 = the_set_of_l1RealSequences & dom NORM2 = the_set_of_l1RealSequences ) by FUNCT_2:def 1;
hence NORM1 = NORM2 by A4, FUNCT_1:2; :: thesis: verum