reconsider K5 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= - (p7 `1) } as closed Subset of (TOP-REAL 2) by JGRAPH_2:47;
reconsider K4 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2 } as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
reconsider K3 = { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `1) <= p7 `2 } as closed Subset of (TOP-REAL 2) by JGRAPH_2:47;
reconsider K2 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1 } as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
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) ) );
set b0 = NonZero (TOP-REAL 2);
defpred S2[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) ) );
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 = (Sq_Circ ") | 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 = (Sq_Circ ") | 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 = (Sq_Circ ") | 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 ) )
set k0 = { p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) ) } ;
assume that
A1: f = (Sq_Circ ") | K0 and
A2: ( B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) ) } ) ; :: thesis: ( f is continuous & K0 is closed )
the carrier of ((TOP-REAL 2) | B0) = B0 by PRE_TOPC:8;
then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1;
{ p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) ) } c= NonZero (TOP-REAL 2) from JGRAPH_3:sch 1();
 then A3: ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by A2, PRE_TOPC:7;
reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch 1();
 A4: (K2 /\ K3) \/ (K4 /\ K5) c= K1
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (K2 /\ K3) \/ (K4 /\ K5) or x in K1 )
assume A5: x in (K2 /\ K3) \/ (K4 /\ K5) ; :: thesis: x in K1
per cases ( x in K2 /\ K3 or x in K4 /\ K5 ) by A5, XBOOLE_0:def 3;
suppose A6: x in K2 /\ K3 ; :: thesis: x in K1
then x in K3 by XBOOLE_0:def 4;
then A7: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & - (p8 `1) <= p8 `2 ) ;
x in K2 by A6, 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 A7; :: thesis: verum
end;
suppose A8: x in K4 /\ K5 ; :: thesis: x in K1
then x in K5 by XBOOLE_0:def 4;
then A9: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & p8 `2 <= - (p8 `1) ) ;
x in K4 by A8, 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 A9; :: thesis: verum
end;
end;
end;
A10: ( K2 /\ K3 is closed & K4 /\ K5 is closed ) by TOPS_1:8;
K1 c= (K2 /\ K3) \/ (K4 /\ K5)
proof
let x be object ; :: 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 A4;
then A11: K1 is closed by A10, TOPS_1:9;
{ p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) ) } = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } /\ (NonZero (TOP-REAL 2)) from JGRAPH_3:sch 2();
 then K0 = K1 /\ ([#] ((TOP-REAL 2) | B0)) by A2, PRE_TOPC:def 5;
hence ( f is continuous & K0 is closed ) by A1, A2, A3, A11, Th37, PRE_TOPC:13; :: thesis: verum