set GG = [:I[01],I[01]:];
set SS = [:R^1,R^1:];
0 in the carrier of I[01] by BORSUK_1:43;
then [0,0] in [: the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
then reconsider x = [0,0] as Point of [:I[01],I[01]:] by BORSUK_1:def 2;
reconsider PA = { p where p is Point of [:R^1,R^1:] : p `2 <= 1 - (2 * (p `1)) } as closed Subset of [:R^1,R^1:] by Th21;
set P0 = { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } ;
A1: [:I[01],I[01]:] is SubSpace of [:R^1,R^1:] by BORSUK_3:21;
A2: { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } = PA /\ ([#] [:I[01],I[01]:])

proof

thus { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `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)) }

proof

let x be object ; :: 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)) } or x in PA /\ ([#] [:I[01],I[01]:]) )
A3: the carrier of [:I[01],I[01]:] c= the carrier of [:R^1,R^1:] by A1, BORSUK_1:1;
assume x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } ; :: thesis: x in PA /\ ([#] [:I[01],I[01]:])
then A4: ex p being Point of [:I[01],I[01]:] st
( x = p & p `2 <= 1 - (2 * (p `1)) ) ;
then x in the carrier of [:I[01],I[01]:] ;
then reconsider a = x as Point of [:R^1,R^1:] by A3;
a `2 <= 1 - (2 * (a `1)) by A4;
then x in PA ;
hence x in PA /\ ([#] [:I[01],I[01]:]) by A4, XBOOLE_0:def 4; :: thesis: verum

end;

let x be object ; :: 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)) } )
assume A5: 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)) }
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)) ) ;
hence x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } by A5; :: thesis: verum

end;

x `2 <= 1 - (2 * (x `1)) ;
then x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } ;
hence { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } is non empty closed Subset of [:I[01],I[01]:] by A1, A2, PRE_TOPC:13; :: thesis: verum