let seq1, seq2 be Real_Sequence; :: thesis: ( seq1 is bounded_above & seq1 is nonnegative & seq2 is bounded_above & seq2 is nonnegative implies ( seq1 (#) seq2 is bounded_above & upper_bound (seq1 (#) seq2) <= (upper_bound seq1) * (upper_bound seq2) ) )
assume that
A1: ( seq1 is bounded_above & seq1 is nonnegative ) and
A2: ( seq2 is bounded_above & seq2 is nonnegative ) ; :: thesis: ( seq1 (#) seq2 is bounded_above & upper_bound (seq1 (#) seq2) <= (upper_bound seq1) * (upper_bound seq2) )
for n being Element of NAT holds (seq1 (#) seq2) . n <= (upper_bound seq1) * (upper_bound seq2)

proof

let n be Element of NAT ; :: thesis: (seq1 (#) seq2) . n <= (upper_bound seq1) * (upper_bound seq2)
A3: ( (seq1 (#) seq2) . n = (seq1 . n) * (seq2 . n) & seq1 . n <= upper_bound seq1 ) by A1, Th7, SEQ_1:8;
A4: seq2 . n >= 0 by A2, Def3;
( seq2 . n <= upper_bound seq2 & seq1 . n >= 0 ) by A1, A2, Def3, Th7;
hence (seq1 (#) seq2) . n <= (upper_bound seq1) * (upper_bound seq2) by A3, A4, XREAL_1:66; :: thesis: verum

end;

hence ( seq1 (#) seq2 is bounded_above & upper_bound (seq1 (#) seq2) <= (upper_bound seq1) * (upper_bound seq2) ) by Th9; :: thesis: verum