let s1, s2, t1, t2 be Real; :: thesis: for P, P1 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } holds
P1 c= [#] ((TOP-REAL 2) | (P `))
let P, P1 be Subset of (TOP-REAL 2); :: thesis: ( P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } implies P1 c= [#] ((TOP-REAL 2) | (P `)) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } and
A2: P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } ; :: thesis: P1 c= [#] ((TOP-REAL 2) | (P `))
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in P1 or x in [#] ((TOP-REAL 2) | (P `)) )
assume A3: x in P1 ; :: thesis: x in [#] ((TOP-REAL 2) | (P `))
then A4: ex pa being Point of (TOP-REAL 2) st
( pa = x & s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) by A2;

now :: thesis: not x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) }

assume x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } ; :: thesis: contradiction
then ex p being Point of (TOP-REAL 2) st
( p = x & ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) ) ;
hence contradiction by A4; :: thesis: verum

end;

then x in P ` by A1, A3, SUBSET_1:29;
hence x in [#] ((TOP-REAL 2) | (P `)) by PRE_TOPC:def 5; :: thesis: verum