let X be set ; :: thesis: for f being Function of [:X,X:],REAL st f is_metric_of X holds
f is_a_pseudometric_of X
let f be Function of [:X,X:],REAL; :: thesis: ( f is_metric_of X implies f is_a_pseudometric_of X )
assume f is_metric_of X ; :: thesis: f is_a_pseudometric_of X
then for a, b, c being Element of X holds
( f . (a,a) = 0 & f . (a,b) = f . (b,a) & f . (a,c) <= (f . (a,b)) + (f . (b,c)) ) by PCOMPS_1:def 6;
then ( f is Reflexive & f is symmetric & f is triangle ) by METRIC_1:def 2, METRIC_1:def 4, METRIC_1:def 5;
hence f is_a_pseudometric_of X by NAGATA_1:def 10; :: thesis: verum