defpred S₁[ set , set ] means for A, B being Element of Closed_Domains_of T st $1 = [A,B] holds
$2 = Cl (Int (A /\ B));
set D = [:(Closed_Domains_of T),(Closed_Domains_of T):];
A1: for a being Element of [:(Closed_Domains_of T),(Closed_Domains_of T):] ex b being Element of Closed_Domains_of T st S₁[a,b]

proof

let a be Element of [:(Closed_Domains_of T),(Closed_Domains_of T):]; :: thesis: ex b being Element of Closed_Domains_of T st S₁[a,b]
reconsider G = a `1 , F = a `2 as Element of Closed_Domains_of T ;
Cl (Int (G /\ F)) is closed_condensed by Th22;
then Cl (Int (G /\ F)) in { E where E is Subset of T : E is closed_condensed } ;
then reconsider b = Cl (Int (G /\ F)) as Element of Closed_Domains_of T ;
take b ; :: thesis: S₁[a,b]
let A, B be Element of Closed_Domains_of T; :: thesis: ( a = [A,B] implies b = Cl (Int (A /\ B)) )
assume a = [A,B] ; :: thesis: b = Cl (Int (A /\ B))
then A2: [A,B] = [G,F] by MCART_1:21;
then A = G by XTUPLE_0:1;
hence b = Cl (Int (A /\ B)) by A2, XTUPLE_0:1; :: thesis: verum

end;

consider h being Function of [:(Closed_Domains_of T),(Closed_Domains_of T):],(Closed_Domains_of T) such that
A3: for a being Element of [:(Closed_Domains_of T),(Closed_Domains_of T):] holds S₁[a,h . a] from FUNCT_2:sch 3(A1);
take h ; :: thesis: for A, B being Element of Closed_Domains_of T holds h . (A,B) = Cl (Int (A /\ B))
let A, B be Element of Closed_Domains_of T; :: thesis: h . (A,B) = Cl (Int (A /\ B))
thus h . (A,B) = h . [A,B]
.= Cl (Int (A /\ B)) by A3 ; :: thesis: verum