set GG = [:I[01] ,I[01] :];
set SS = [:R^1 ,R^1 :];
1 in the carrier of I[01] by BORSUK_1:86;
then [1,1] in [:the carrier of I[01] ,the carrier of I[01] :] by ZFMISC_1:106;
then reconsider x = [1,1] as Point of [:I[01] ,I[01] :] by BORSUK_1:def 5;
reconsider PA = { p where p is Point of [:R^1 ,R^1 :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } as closed Subset of [:R^1 ,R^1 :] by Th27;
set P0 = { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } ;
A1: x `1 = 1 by MCART_1:7;
then A2: x `2 >= (2 * (x `1 )) - 1 by MCART_1:7;
A3: [:I[01] ,I[01] :] is SubSpace of [:R^1 ,R^1 :] by BORSUK_3:25;
A4: { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } = PA /\ ([#] [:I[01] ,I[01] :])

proof

thus { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } c= PA /\ ([#] [:I[01] ,I[01] :]) :: according to XBOOLE_0:def 10 :: thesis: PA /\ ([#] [:I[01] ,I[01] :]) c= { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) }

proof

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 PA /\ ([#] [:I[01] ,I[01] :]) )
A5: the carrier of [:I[01] ,I[01] :] c= the carrier of [:R^1 ,R^1 :] by A3, BORSUK_1:35;
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 PA /\ ([#] [:I[01] ,I[01] :])
then A6: ex p being Point of [:I[01] ,I[01] :] st
( x = p & p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) ;
then x in the carrier of [:I[01] ,I[01] :] ;
then reconsider a = x as Point of [:R^1 ,R^1 :] by A5;
a `2 >= 1 - (2 * (a `1 )) by A6;
then x in PA by A6;
hence x in PA /\ ([#] [:I[01] ,I[01] :]) by A6, XBOOLE_0:def 4; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in PA /\ ([#] [:I[01] ,I[01] :]) 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 A7: x in PA /\ ([#] [:I[01] ,I[01] :]) ; :: 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 x in PA by XBOOLE_0:def 4;
then ex p being Point of [:R^1 ,R^1 :] st
( x = p & p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) ;
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 A7; :: thesis: verum

end;

x `2 = 1 by MCART_1:7;
then x `2 >= 1 - (2 * (x `1 )) by A1;
then 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;
hence { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } is non empty closed Subset of [:I[01] ,I[01] :] by A3, A4, PRE_TOPC:43; :: thesis: verum