:: deftheorem defines is_lower_semicontinuous_on RFUNCT_4:def 8 :
for f being PartFunc of REAL,REAL
for X being set holds
( f is_lower_semicontinuous_on X iff ( X c= dom f & ( for x0 being Real st x0 in X holds
f | X is_lower_semicontinuous_in x0 ) ) );