consider a being set ;
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 non empty Subset of X
for x being Element of X 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 non empty Subset of X
for x being Element of X holds (lower_bound f,A) . x >= 0 )
assume A1: f is_a_pseudometric_of X ; :: thesis: for A being non empty Subset of X
for x being Element of X holds (lower_bound f,A) . x >= 0
let A be non empty Subset of X; :: thesis: for x being Element of X holds (lower_bound f,A) . x >= 0
let x be Element of X; :: thesis: (lower_bound f,A) . x >= 0

A2: now

let rn be real number ; :: thesis: ( rn in (dist f,x) .: A implies rn >= 0 )
assume rn in (dist f,x) .: A ; :: thesis: rn >= 0
then consider y being set such that
A3: y in dom (dist f,x) and
y in A and
A4: rn = (dist f,x) . y by FUNCT_1:def 12;
(dist f,x) . y = f . x,y by A3, Def2;
hence rn >= 0 by A1, A3, A4, NAGATA_1:29; :: thesis: verum

end;

X = dom (dist f,x) by FUNCT_2:def 1;
then lower_bound ((dist f,x) .: A) >= 0 by A2, SEQ_4:60;
hence (lower_bound f,A) . x >= 0 by Def3; :: thesis: verum