let seq1, seq2 be Real_Sequence; :: thesis: ( seq1 is bounded_below & seq1 is nonnegative & seq2 is bounded_below & seq2 is nonnegative implies ( seq1 (#) seq2 is bounded_below & inf (seq1 (#) seq2) >= (inf seq1) * (inf seq2) ) )
assume A1: ( seq1 is bounded_below & seq1 is nonnegative & seq2 is bounded_below & seq2 is nonnegative ) ; :: thesis: ( seq1 (#) seq2 is bounded_below & inf (seq1 (#) seq2) >= (inf seq1) * (inf seq2) )
for n being Element of NAT holds (seq1 (#) seq2) . n >= (inf seq1) * (inf seq2)

proof

let n be Element of NAT ; :: thesis: (seq1 (#) seq2) . n >= (inf seq1) * (inf seq2)
A2: (seq1 (#) seq2) . n = (seq1 . n) * (seq2 . n) by SEQ_1:12;
A3: ( seq1 . n >= inf seq1 & seq2 . n >= inf seq2 ) by A1, Th8;
( inf seq1 >= 0 & inf seq2 >= 0 ) by A1, Th18;
hence (seq1 (#) seq2) . n >= (inf seq1) * (inf seq2) by A2, A3, XREAL_1:68; :: thesis: verum

end;

hence ( seq1 (#) seq2 is bounded_below & inf (seq1 (#) seq2) >= (inf seq1) * (inf seq2) ) by Th10; :: thesis: verum