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

proof

let a be Element of [:(Open_Domains_of T),(Open_Domains_of T):]; :: thesis: ex b being Element of Open_Domains_of T st S₁[a,b]
reconsider G = a `1 , F = a `2 as Element of Open_Domains_of T ;
G in { E where E is Subset of T : E is open_condensed } ;
then consider E being Subset of T such that
A2: E = G and
A3: E is open_condensed ;
F in { H where H is Subset of T : H is open_condensed } ;
then consider H being Subset of T such that
A4: H = F and
A5: H is open_condensed ;
E /\ H is open_condensed by A3, A5, TOPS_1:69;
then G /\ F in { K where K is Subset of T : K is open_condensed } by A2, A4;
then reconsider b = G /\ F as Element of Open_Domains_of T ;
take b ; :: thesis: S₁[a,b]
let A, B be Element of Open_Domains_of T; :: thesis: ( a = [A,B] implies b = A /\ B )
assume a = [A,B] ; :: thesis: b = A /\ B
then A6: [A,B] = [G,F] by MCART_1:21;
then A = G by XTUPLE_0:1;
hence b = A /\ B by A6, XTUPLE_0:1; :: thesis: verum

end;

consider h being Function of [:(Open_Domains_of T),(Open_Domains_of T):],(Open_Domains_of T) such that
A7: for a being Element of [:(Open_Domains_of T),(Open_Domains_of T):] holds S₁[a,h . a] from FUNCT_2:sch 3(A1);
take h ; :: thesis: for A, B being Element of Open_Domains_of T holds h . (A,B) = A /\ B
let A, B be Element of Open_Domains_of T; :: thesis: h . (A,B) = A /\ B
thus h . (A,B) = h . [A,B]
.= A /\ B by A7 ; :: thesis: verum