let X be non empty set ; :: thesis: for f being Function of [:X,X:],REAL st f is_a_pseudometric_of X holds
for A being Subset of X
for x being Element of X st x in A holds
(lower_bound f,A) . x = 0
let f be Function of [:X,X:],REAL ; :: thesis: ( f is_a_pseudometric_of X implies for A being Subset of X
for x being Element of X st x in A holds
(lower_bound f,A) . x = 0 )
assume A1: f is_a_pseudometric_of X ; :: thesis: for A being Subset of X
for x being Element of X st x in A holds
(lower_bound f,A) . x = 0
let A be Subset of X; :: thesis: for x being Element of X st x in A holds
(lower_bound f,A) . x = 0
let x be Element of X; :: thesis: ( x in A implies (lower_bound f,A) . x = 0 )
assume A2: x in A ; :: thesis: (lower_bound f,A) . x = 0
then reconsider A = A as non empty Subset of X ;
A3: ( not (dist f,x) .: A is empty & (dist f,x) .: A is bounded_below ) by A1, Lm1;
f is Reflexive by A1, NAGATA_1:def 10;
then f . x,x = 0 by METRIC_1:def 3;
then ( X = dom (dist f,x) & (dist f,x) . x = 0 ) by Def2, FUNCT_2:def 1;
then 0 in (dist f,x) .: A by A2, FUNCT_1:def 12;
then lower_bound ((dist f,x) .: A) <= 0 by A3, SEQ_4:def 5;
then (lower_bound f,A) . x <= 0 by Def3;
hence (lower_bound f,A) . x = 0 by A1, Th5; :: thesis: verum