set P = { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } ;
thus IBB c= { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } :: according to XBOOLE_0:def 10 :: thesis: { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } c= IBB

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in IBB or x in { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } )
assume A1: x in IBB ; :: thesis: x in { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) }
then consider a, b being Point of I[01] such that
A2: ( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) by Def11;
reconsider x' = x as Point of [:I[01] ,I[01] :] by A1;
( x' `1 = a & x' `2 = b ) by A2, MCART_1:7;
hence x in { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } by A2; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } or x in IBB )
assume x in { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } ; :: thesis: x in IBB
then consider p being Point of [:I[01] ,I[01] :] such that
A3: ( p = x & p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) ;
x in the carrier of [:I[01] ,I[01] :] by A3;
then x in [:the carrier of I[01] ,the carrier of I[01] :] by BORSUK_1:def 5;
then A4: x = [(x `1 ),(x `2 )] by MCART_1:23;
A5: p `1 is Point of I[01] by Th31;
p `2 is Point of I[01] by Th31;
hence x in IBB by A3, A4, A5, Def11; :: thesis: verum