let X be set ; :: thesis: for f being Function of [:X,X:],REAL holds
( f is_a_pseudometric_of X iff 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) ) )
let f be Function of [:X,X:],REAL ; :: thesis: ( f is_a_pseudometric_of X iff 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) ) )
( f is_a_pseudometric_of X iff ( f is Reflexive & f is symmetric & f is triangle ) ) by Def10;
hence ( f is_a_pseudometric_of X iff 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 METRIC_1:def 3, METRIC_1:def 5, METRIC_1:def 6; :: thesis: verum