consider r1 being Real such that
A1: for c being object st c in dom f holds
f . c < r1 by SEQ_2:def 1;

now :: thesis: ex p being set st
for c being object st c in dom (r (#) f) holds
p < (r (#) f) . c

take p = (r * |.r1.|) - 1; :: thesis: for c being object st c in dom (r (#) f) holds
p < (r (#) f) . c
let c be object ; :: thesis: ( c in dom (r (#) f) implies p < (r (#) f) . c )
A2: r1 <= |.r1.| by ABSVALUE:4;
assume A3: c in dom (r (#) f) ; :: thesis: p < (r (#) f) . c
then c in dom f by VALUED_1:def 5;
then f . c <= |.r1.| by A1, A2, XXREAL_0:2;
then r * |.r1.| <= r * (f . c) by XREAL_1:65;
then r * |.r1.| <= (r (#) f) . c by A3, VALUED_1:def 5;
hence p < (r (#) f) . c by XREAL_1:231; :: thesis: verum

end;

hence for b₁ being real-valued Function st b₁ = r (#) f holds
b₁ is bounded_below ; :: thesis: verum