reconsider C = [:(NetUniv T), the carrier of T:] as Convergence-Class of T by Def21;
take C ; :: thesis: ( not C is empty & C is topological )
thus not C is empty ; :: thesis: C is topological
thus C is topological :: thesis: verum

proof

thus C is (CONSTANTS) :: according to YELLOW_6:def 25 :: thesis: ( C is (SUBNETS) & C is (DIVERGENCE) & C is (ITERATED_LIMITS) )

proof

let N be constant net of T; :: according to YELLOW_6:def 20 :: thesis: ( N in NetUniv T implies [N,(the_value_of N)] in C )
thus ( N in NetUniv T implies [N,(the_value_of N)] in C ) by ZFMISC_1:87; :: thesis: verum

end;

thus C is (SUBNETS) :: thesis: ( C is (DIVERGENCE) & C is (ITERATED_LIMITS) )

proof

let N be net of T; :: according to YELLOW_6:def 21 :: thesis: for Y being subnet of N st Y in NetUniv T holds
for p being Element of T st [N,p] in C holds
[Y,p] in C
let Y be subnet of N; :: thesis: ( Y in NetUniv T implies for p being Element of T st [N,p] in C holds
[Y,p] in C )
thus ( Y in NetUniv T implies for p being Element of T st [N,p] in C holds
[Y,p] in C ) by ZFMISC_1:87; :: thesis: verum

end;

thus C is (DIVERGENCE) :: thesis: C is (ITERATED_LIMITS)

proof

let X be net of T; :: according to YELLOW_6:def 22 :: thesis: for p being Element of T st X in NetUniv T & not [X,p] in C holds
ex Y being subnet of X st
( Y in NetUniv T & ( for Z being subnet of Y holds not [Z,p] in C ) )
let p be Element of T; :: thesis: ( X in NetUniv T & not [X,p] in C implies ex Y being subnet of X st
( Y in NetUniv T & ( for Z being subnet of Y holds not [Z,p] in C ) ) )
thus ( X in NetUniv T & not [X,p] in C implies ex Y being subnet of X st
( Y in NetUniv T & ( for Z being subnet of Y holds not [Z,p] in C ) ) ) by ZFMISC_1:87; :: thesis: verum

end;

let X be net of T; :: according to YELLOW_6:def 23 :: thesis: for p being Element of T st [X,p] in C holds
for J being net_set of the carrier of X,T st ( for i being Element of X holds [(J . i),(X . i)] in C ) holds
[(Iterated J),p] in C
let p be Element of T; :: thesis: ( [X,p] in C implies for J being net_set of the carrier of X,T st ( for i being Element of X holds [(J . i),(X . i)] in C ) holds
[(Iterated J),p] in C )
assume [X,p] in C ; :: thesis: for J being net_set of the carrier of X,T st ( for i being Element of X holds [(J . i),(X . i)] in C ) holds
[(Iterated J),p] in C
then A1: X in NetUniv T by ZFMISC_1:87;
let J be net_set of the carrier of X,T; :: thesis: ( ( for i being Element of X holds [(J . i),(X . i)] in C ) implies [(Iterated J),p] in C )
assume A2: for i being Element of X holds [(J . i),(X . i)] in C ; :: thesis: [(Iterated J),p] in C

now

let i be Element of X; :: thesis: J . i in NetUniv T
[(J . i),(X . i)] in C by A2;
hence J . i in NetUniv T by ZFMISC_1:87; :: thesis: verum

end;

then Iterated J in NetUniv T by A1, Th34;
hence [(Iterated J),p] in C by ZFMISC_1:87; :: thesis: verum

end;