let X, Y be non empty TopSpace; :: thesis: for D being Subset of [:X,Y:] st D is open holds
for X1 being Subset of X
for Y1 being Subset of Y holds
( ( X1 = (pr1 ( the carrier of X, the carrier of Y)) .: D implies X1 is open ) & ( Y1 = (pr2 ( the carrier of X, the carrier of Y)) .: D implies Y1 is open ) )
let D be Subset of [:X,Y:]; :: thesis: ( D is open implies for X1 being Subset of X
for Y1 being Subset of Y holds
( ( X1 = (pr1 ( the carrier of X, the carrier of Y)) .: D implies X1 is open ) & ( Y1 = (pr2 ( the carrier of X, the carrier of Y)) .: D implies Y1 is open ) ) )
set p = pr2 ( the carrier of X, the carrier of Y);
set P = .: (pr2 ( the carrier of X, the carrier of Y));
assume D is open ; :: thesis: for X1 being Subset of X
for Y1 being Subset of Y holds
( ( X1 = (pr1 ( the carrier of X, the carrier of Y)) .: D implies X1 is open ) & ( Y1 = (pr2 ( the carrier of X, the carrier of Y)) .: D implies Y1 is open ) )
then consider H being Subset-Family of [:X,Y:] such that
A1: D = union H and
A2: for e being set st e in H holds
ex X1 being Subset of X ex Y1 being Subset of Y st
( e = [:X1,Y1:] & X1 is open & Y1 is open ) by Th5;
reconsider K = H as Subset-Family of [: the carrier of X, the carrier of Y:] by Def2;
let X1 be Subset of X; :: thesis: for Y1 being Subset of Y holds
( ( X1 = (pr1 ( the carrier of X, the carrier of Y)) .: D implies X1 is open ) & ( Y1 = (pr2 ( the carrier of X, the carrier of Y)) .: D implies Y1 is open ) )
let Y1 be Subset of Y; :: thesis: ( ( X1 = (pr1 ( the carrier of X, the carrier of Y)) .: D implies X1 is open ) & ( Y1 = (pr2 ( the carrier of X, the carrier of Y)) .: D implies Y1 is open ) )
thus ( X1 = (pr1 ( the carrier of X, the carrier of Y)) .: D implies X1 is open ) :: thesis: ( Y1 = (pr2 ( the carrier of X, the carrier of Y)) .: D implies Y1 is open )

proof

set p = pr1 ( the carrier of X, the carrier of Y);
set P = .: (pr1 ( the carrier of X, the carrier of Y));
reconsider PK = (.: (pr1 ( the carrier of X, the carrier of Y))) .: K as Subset-Family of X ;
reconsider PK = PK as Subset-Family of X ;
A3: PK is open

proof

let X1 be Subset of X; :: according to TOPS_2:def 1 :: thesis: ( not X1 in PK or X1 is open )
reconsider x1 = X1 as Subset of X ;
assume X1 in PK ; :: thesis: X1 is open
then consider c being Subset of [: the carrier of X, the carrier of Y:] such that
A4: c in K and
A5: x1 = (.: (pr1 ( the carrier of X, the carrier of Y))) . c by FUNCT_2:65;
dom (.: (pr1 ( the carrier of X, the carrier of Y))) = bool [: the carrier of X, the carrier of Y:] by FUNCT_2:def 1;
then A6: X1 = (pr1 ( the carrier of X, the carrier of Y)) .: c by A5, FUNCT_3:7;
A7: ( c = {} or c <> {} ) ;
ex X2 being Subset of X ex Y2 being Subset of Y st
( c = [:X2,Y2:] & X2 is open & Y2 is open ) by A2, A4;
hence X1 is open by A6, A7, EQREL_1:49; :: thesis: verum

end;

assume X1 = (pr1 ( the carrier of X, the carrier of Y)) .: D ; :: thesis: X1 is open
then X1 = union ((.: (pr1 ( the carrier of X, the carrier of Y))) .: K) by A1, EQREL_1:53;
hence X1 is open by A3, TOPS_2:19; :: thesis: verum

end;

reconsider PK = (.: (pr2 ( the carrier of X, the carrier of Y))) .: K as Subset-Family of Y ;
reconsider PK = PK as Subset-Family of Y ;
A8: PK is open

proof

let Y1 be Subset of Y; :: according to TOPS_2:def 1 :: thesis: ( not Y1 in PK or Y1 is open )
reconsider y1 = Y1 as Subset of Y ;
assume Y1 in PK ; :: thesis: Y1 is open
then consider c being Subset of [: the carrier of X, the carrier of Y:] such that
A9: c in K and
A10: y1 = (.: (pr2 ( the carrier of X, the carrier of Y))) . c by FUNCT_2:65;
dom (.: (pr2 ( the carrier of X, the carrier of Y))) = bool [: the carrier of X, the carrier of Y:] by FUNCT_2:def 1;
then A11: Y1 = (pr2 ( the carrier of X, the carrier of Y)) .: c by A10, FUNCT_3:7;
A12: ( c = {} or c <> {} ) ;
ex X2 being Subset of X ex Y2 being Subset of Y st
( c = [:X2,Y2:] & X2 is open & Y2 is open ) by A2, A9;
hence Y1 is open by A11, A12, EQREL_1:49; :: thesis: verum

end;

assume Y1 = (pr2 ( the carrier of X, the carrier of Y)) .: D ; :: thesis: Y1 is open
then Y1 = union ((.: (pr2 ( the carrier of X, the carrier of Y))) .: K) by A1, EQREL_1:53;
hence Y1 is open by A8, TOPS_2:19; :: thesis: verum