for p being Point of T
for V being Subset of Y st (Prj2 s,H) . p in V & V is open holds
ex W being Subset of T st
( p in W & W is open & (Prj2 s,H) .: W c= V )

let p be Point of T; :: thesis:
let V be Subset of Y; :: thesis:
assume A1: ( (Prj2 s,H) . p in V & V is open ) ; :: thesis:
(Prj2 s,H) . p = H . s,p by Def3;
then consider W being Subset of [:S,T:] such that
A2: [s,p] in W and
A3: W is open and
A4: H .: W c= V by A1, JGRAPH_2:20;
consider A being Subset-Family of [:S,T:] such that
A5: W = union A and
A6: for e being set st e in A holds
ex X1 being Subset of S ex Y1 being Subset of T st
( e = [:X1,Y1:] & X1 is open & Y1 is open ) by A3, BORSUK_1:45;
consider e being set such that
A7: [s,p] in e and
A8: e in A by A2, A5, TARSKI:def 4;
consider X1 being Subset of S, Y1 being Subset of T such that
A9: e = [:X1,Y1:] and
X1 is open and
A10: Y1 is open by A6, A8;
take Y1 ; :: thesis:
thus p in Y1 by A7, A9, ZFMISC_1:106; :: thesis:
thus Y1 is open by A10; :: thesis:
let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in (Prj2 s,H) .: Y1 ; :: thesis:
then consider c being Point of T such that
A11: c in Y1 and
A12: x = (Prj2 s,H) . c by FUNCT_2:116;
s in X1 by A7, A9, ZFMISC_1:106;
then [s,c] in [:X1,Y1:] by A11, ZFMISC_1:106;
then [s,c] in W by A5, A8, A9, TARSKI:def 4;
then A13: H . [s,c] in H .: W by FUNCT_2:43;
(Prj2 s,H) . c = H . s,c by Def3
.= H . [s,c] ;
hence x in V by A4, A12, A13; :: thesis:

end;