now :: thesis: ex p being Real st
for c being object st c in dom (abs f) holds
p < (abs f) . c

reconsider p = - 1 as Real ;
take p = p; :: thesis: for c being object st c in dom (abs f) holds
p < (abs f) . c
let c be object ; :: thesis: ( c in dom (abs f) implies p < (abs f) . c )
assume c in dom (abs f) ; :: thesis: p < (abs f) . c
0 <= |.(f . c).| by COMPLEX1:46;
hence p < (abs f) . c by VALUED_1:18; :: thesis: verum

end;

hence abs f is bounded_below ; :: thesis: verum