reconsider K5 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= - (p7 `1 ) } as closed Subset of (TOP-REAL 2) by Lm14;
reconsider K4 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2 } as closed Subset of (TOP-REAL 2) by Lm5;
reconsider K3 = { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `1 ) <= p7 `2 } as closed Subset of (TOP-REAL 2) by Lm11;
reconsider K2 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1 } as closed Subset of (TOP-REAL 2) by Lm8;
let B0 be Subset of (TOP-REAL 2); :: thesis: for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )
let K0 be Subset of ((TOP-REAL 2) | B0); :: thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )
let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); :: thesis: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0. (TOP-REAL 2) ) } implies ( f is continuous & K0 is closed ) )
defpred S₁[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1 ) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1 ) ) );
A1: ( K2 /\ K3 is closed & K4 /\ K5 is closed ) by TOPS_1:35;
the carrier of ((TOP-REAL 2) | B0) = [#] ((TOP-REAL 2) | B0)
.= B0 by PRE_TOPC:def 10 ;
then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1;
assume A2: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0. (TOP-REAL 2) ) } ) ; :: thesis: ( f is continuous & K0 is closed )
K0 c= B0

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K0 or x in B0 )
assume x in K0 ; :: thesis: x in B0
then A3: ex p8 being Point of (TOP-REAL 2) st
( x = p8 & ( ( p8 `2 <= p8 `1 & - (p8 `1 ) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1 ) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def 1;
hence x in B0 by A2, A3, XBOOLE_0:def 5; :: thesis: verum

end;

then A4: ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:28;
reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S₁[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch 1();
A5: K1 /\ B0 c= K0

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K1 /\ B0 or x in K0 )
assume A6: x in K1 /\ B0 ; :: thesis: x in K0
then x in B0 by XBOOLE_0:def 4;
then not x in {(0. (TOP-REAL 2))} by A2, XBOOLE_0:def 5;
then A7: not x = 0. (TOP-REAL 2) by TARSKI:def 1;
x in K1 by A6, XBOOLE_0:def 4;
then ex p7 being Point of (TOP-REAL 2) st
( p7 = x & ( ( p7 `2 <= p7 `1 & - (p7 `1 ) <= p7 `2 ) or ( p7 `2 >= p7 `1 & p7 `2 <= - (p7 `1 ) ) ) ) ;
hence x in K0 by A2, A7; :: thesis: verum

end;

A8: (K2 /\ K3) \/ (K4 /\ K5) c= K1

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (K2 /\ K3) \/ (K4 /\ K5) or x in K1 )
assume A9: x in (K2 /\ K3) \/ (K4 /\ K5) ; :: thesis: x in K1

now

per cases ( x in K2 /\ K3 or x in K4 /\ K5 ) by A9, XBOOLE_0:def 3;

case A10: x in K2 /\ K3 ; :: thesis: x in K1

then x in K3 by XBOOLE_0:def 4;
then A11: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & - (p8 `1 ) <= p8 `2 ) ;
x in K2 by A10, XBOOLE_0:def 4;
then ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `2 <= p7 `1 ) ;
hence x in K1 by A11; :: thesis: verum

end;

case A12: x in K4 /\ K5 ; :: thesis: x in K1

then x in K5 by XBOOLE_0:def 4;
then A13: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & p8 `2 <= - (p8 `1 ) ) ;
x in K4 by A12, XBOOLE_0:def 4;
then ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `2 >= p7 `1 ) ;
hence x in K1 by A13; :: thesis: verum

end;

end;

end;

hence x in K1 ; :: thesis: verum

end;

K1 c= (K2 /\ K3) \/ (K4 /\ K5)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K1 or x in (K2 /\ K3) \/ (K4 /\ K5) )
assume x in K1 ; :: thesis: x in (K2 /\ K3) \/ (K4 /\ K5)
then ex p being Point of (TOP-REAL 2) st
( p = x & ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) ) ;
then ( ( x in K2 & x in K3 ) or ( x in K4 & x in K5 ) ) ;
then ( x in K2 /\ K3 or x in K4 /\ K5 ) by XBOOLE_0:def 4;
hence x in (K2 /\ K3) \/ (K4 /\ K5) by XBOOLE_0:def 3; :: thesis: verum

end;

then K1 = (K2 /\ K3) \/ (K4 /\ K5) by A8, XBOOLE_0:def 10;
then A14: K1 is closed by A1, TOPS_1:36;
K0 c= K1 /\ B0

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K0 or x in K1 /\ B0 )
assume x in K0 ; :: thesis: x in K1 /\ B0
then A15: ex p being Point of (TOP-REAL 2) st
( x = p & ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0. (TOP-REAL 2) ) by A2;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def 1;
then A16: x in B0 by A2, A15, XBOOLE_0:def 5;
x in K1 by A15;
hence x in K1 /\ B0 by A16, XBOOLE_0:def 4; :: thesis: verum

end;

then K0 = K1 /\ B0 by A5, XBOOLE_0:def 10
.= K1 /\ ([#] ((TOP-REAL 2) | B0)) by PRE_TOPC:def 10 ;
hence ( f is continuous & K0 is closed ) by A2, A4, A14, Th46, PRE_TOPC:43; :: thesis: verum