let sn be Real; :: thesis: for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW ) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous
let K0, B0 be Subset of (TOP-REAL 2); :: thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW ) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous
let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); :: thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW ) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous )
set cn = sqrt (1 - (sn ^2 ));
set p0 = |[(- (sqrt (1 - (sn ^2 )))),sn]|;
A1: |[(- (sqrt (1 - (sn ^2 )))),sn]| `1 = - (sqrt (1 - (sn ^2 ))) by EUCLID:56;
|[(- (sqrt (1 - (sn ^2 )))),sn]| `2 = sn by EUCLID:56;
then A2: |.|[(- (sqrt (1 - (sn ^2 )))),sn]|.| = sqrt (((- (sqrt (1 - (sn ^2 )))) ^2 ) + (sn ^2 )) by A1, JGRAPH_3:10
.= sqrt (((sqrt (1 - (sn ^2 ))) ^2 ) + (sn ^2 )) ;
assume A3: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW ) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; :: thesis: f is continuous
then sn ^2 < 1 ^2 by SQUARE_1:120;
then A4: 1 - (sn ^2 ) > 0 by XREAL_1:52;
then A5: - (- (sqrt (1 - (sn ^2 )))) > 0 by SQUARE_1:93;
A6: now
assume |[(- (sqrt (1 - (sn ^2 )))),sn]| = 0. (TOP-REAL 2) ; :: thesis: contradiction
then - (- (sqrt (1 - (sn ^2 )))) = - 0 by EUCLID:56, JGRAPH_2:11;
hence contradiction by A4, SQUARE_1:93; :: thesis: verum
end;
then |[(- (sqrt (1 - (sn ^2 )))),sn]| in K0 by A3, A1, A5;
then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ;
(sqrt (1 - (sn ^2 ))) ^2 = 1 - (sn ^2 ) by A4, SQUARE_1:def 4;
then A7: (|[(- (sqrt (1 - (sn ^2 )))),sn]| `2 ) / |.|[(- (sqrt (1 - (sn ^2 )))),sn]|.| = sn by A2, EUCLID:56, SQUARE_1:83;
then A8: |[(- (sqrt (1 - (sn ^2 )))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } by A1, A6, A5;
not |[(- (sqrt (1 - (sn ^2 )))),sn]| in {(0. (TOP-REAL 2))} by A6, TARSKI:def 1;
then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A3, XBOOLE_0:def 5;
K1 c= D
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K1 or x in D )
assume A9: x in K1 ; :: thesis: x in D
then ex p6 being Point of (TOP-REAL 2) st
( p6 = x & p6 `1 <= 0 & p6 <> 0. (TOP-REAL 2) ) by A3;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def 1;
hence x in D by A3, A9, XBOOLE_0:def 5; :: thesis: verum
end;
then D = K1 \/ D by XBOOLE_1:12;
then A10: (TOP-REAL 2) | K1 is SubSpace of (TOP-REAL 2) | D by TOPMETR:5;
A11: { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } c= K1
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 )
assume x in { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ; :: thesis: x in K1
then ex p being Point of (TOP-REAL 2) st
( p = x & (p `2 ) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) ;
hence x in K1 by A3; :: thesis: verum
end;
A12: { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } c= K1
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 )
assume x in { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ; :: thesis: x in K1
then ex p being Point of (TOP-REAL 2) st
( p = x & (p `2 ) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) ;
hence x in K1 by A3; :: thesis: verum
end;
then reconsider K00 = { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A8, PRE_TOPC:29;
the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:29;
then A13: rng (f | K00) c= D ;
|[(- (sqrt (1 - (sn ^2 )))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } by A1, A6, A5, A7;
then reconsider K11 = { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A11, PRE_TOPC:29;
the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:29;
then A14: rng (f | K11) c= D ;
the carrier of ((TOP-REAL 2) | B0) = the carrier of ((TOP-REAL 2) | D) ;
then A15: dom f = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def 1
.= K1 by PRE_TOPC:29 ;
then dom (f | K00) = K00 by A12, RELAT_1:91
.= the carrier of (((TOP-REAL 2) | K1) | K00) by PRE_TOPC:29 ;
then reconsider f1 = f | K00 as Function of (((TOP-REAL 2) | K1) | K00),((TOP-REAL 2) | D) by A13, FUNCT_2:4;
dom (f | K11) = K11 by A11, A15, RELAT_1:91
.= the carrier of (((TOP-REAL 2) | K1) | K11) by PRE_TOPC:29 ;
then reconsider f2 = f | K11 as Function of (((TOP-REAL 2) | K1) | K11),((TOP-REAL 2) | D) by A14, FUNCT_2:4;
A16: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
defpred S1[ Point of (TOP-REAL 2)] means ( ($1 `2 ) / |.$1.| >= sn & $1 `1 <= 0 & $1 <> 0. (TOP-REAL 2) );
A17: dom f2 = the carrier of (((TOP-REAL 2) | K1) | K11) by FUNCT_2:def 1
.= K11 by PRE_TOPC:29 ;
{ p where p is Point of (TOP-REAL 2) : S1[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch 7();
 then reconsider K001 = { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A8;
A18: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
defpred S2[ Point of (TOP-REAL 2)] means ( $1 `2 >= sn * |.$1.| & $1 `1 <= 0 );
{ p where p is Point of (TOP-REAL 2) : S2[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch 7();
 then reconsider K003 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 <= 0 ) } as Subset of (TOP-REAL 2) ;
defpred S3[ Point of (TOP-REAL 2)] means ( ($1 `2 ) / |.$1.| <= sn & $1 `1 <= 0 & $1 <> 0. (TOP-REAL 2) );
A19: { p where p is Point of (TOP-REAL 2) : S3[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch 7();
 A20: rng ((sn -FanMorphW ) | K001) c= K1
proof
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ((sn -FanMorphW ) | K001) or y in K1 )
assume y in rng ((sn -FanMorphW ) | K001) ; :: thesis: y in K1
then consider x being set such that
A21: x in dom ((sn -FanMorphW ) | K001) and
A22: y = ((sn -FanMorphW ) | K001) . x by FUNCT_1:def 5;
reconsider q = x as Point of (TOP-REAL 2) by A21;
A23: y = (sn -FanMorphW ) . q by A21, A22, FUNCT_1:70;
dom ((sn -FanMorphW ) | K001) = (dom (sn -FanMorphW )) /\ K001 by RELAT_1:90
.= the carrier of (TOP-REAL 2) /\ K001 by FUNCT_2:def 1
.= K001 by XBOOLE_1:28 ;
then A24: ex p2 being Point of (TOP-REAL 2) st
( p2 = q & (p2 `2 ) / |.p2.| >= sn & p2 `1 <= 0 & p2 <> 0. (TOP-REAL 2) ) by A21;
then A25: ((q `2 ) / |.q.|) - sn >= 0 by XREAL_1:50;
|.q.| <> 0 by A24, TOPRNS_1:25;
then A26: |.q.| ^2 > 0 ^2 by SQUARE_1:74;
set q4 = |[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|;
A27: |[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| `2 = |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)) by EUCLID:56;
A28: 1 - sn > 0 by A3, XREAL_1:151;
0 <= (q `1 ) ^2 by XREAL_1:65;
then 0 + ((q `2 ) ^2 ) <= ((q `1 ) ^2 ) + ((q `2 ) ^2 ) by XREAL_1:9;
then (q `2 ) ^2 <= |.q.| ^2 by JGRAPH_3:10;
then ((q `2 ) ^2 ) / (|.q.| ^2 ) <= (|.q.| ^2 ) / (|.q.| ^2 ) by XREAL_1:74;
then ((q `2 ) ^2 ) / (|.q.| ^2 ) <= 1 by A26, XCMPLX_1:60;
then ((q `2 ) / |.q.|) ^2 <= 1 by XCMPLX_1:77;
then 1 >= (q `2 ) / |.q.| by SQUARE_1:121;
then 1 - sn >= ((q `2 ) / |.q.|) - sn by XREAL_1:11;
then - (1 - sn) <= - (((q `2 ) / |.q.|) - sn) by XREAL_1:26;
then (- (1 - sn)) / (1 - sn) <= (- (((q `2 ) / |.q.|) - sn)) / (1 - sn) by A28, XREAL_1:74;
then - 1 <= (- (((q `2 ) / |.q.|) - sn)) / (1 - sn) by A28, XCMPLX_1:198;
then ((- (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A28, A25, SQUARE_1:119;
then A29: 1 - (((- (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) >= 0 by XREAL_1:50;
then A30: 1 - ((- ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 ) >= 0 by XCMPLX_1:188;
sqrt (1 - (((- (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by A29, SQUARE_1:def 4;
then sqrt (1 - (((- (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 - sn) ^2 ))) >= 0 by XCMPLX_1:77;
then sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 - sn) ^2 ))) >= 0 ;
then A31: sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )) >= 0 by XCMPLX_1:77;
A32: |[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| `1 = |.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) by EUCLID:56;
then A33: (|[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| `1 ) ^2 = (|.q.| ^2 ) * ((sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))) ^2 )
.= (|.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )) by A30, SQUARE_1:def 4 ;
|.|[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|.| ^2 = ((|[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| `1 ) ^2 ) + ((|[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| `2 ) ^2 ) by JGRAPH_3:10
.= |.q.| ^2 by A27, A33 ;
then A34: |[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| <> 0. (TOP-REAL 2) by A26, TOPRNS_1:24;
(sn -FanMorphW ) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A3, A24, Th25;
hence y in K1 by A3, A23, A32, A31, A34; :: thesis: verum
end;
A35: dom (sn -FanMorphW ) = the carrier of (TOP-REAL 2) by FUNCT_2:def 1;
then dom ((sn -FanMorphW ) | K001) = K001 by RELAT_1:91
.= the carrier of ((TOP-REAL 2) | K001) by PRE_TOPC:29 ;
then reconsider f3 = (sn -FanMorphW ) | K001 as Function of ((TOP-REAL 2) | K001),((TOP-REAL 2) | K1) by A18, A20, FUNCT_2:4;
A36: K003 is closed by Th32;
defpred S4[ Point of (TOP-REAL 2)] means ( $1 `2 <= sn * |.$1.| & $1 `1 <= 0 );
{ p where p is Point of (TOP-REAL 2) : S4[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch 7();
 then reconsider K004 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 <= 0 ) } as Subset of (TOP-REAL 2) ;
A37: K004 /\ K1 c= K11
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K004 /\ K1 or x in K11 )
assume A38: x in K004 /\ K1 ; :: thesis: x in K11
then x in K004 by XBOOLE_0:def 4;
then consider q1 being Point of (TOP-REAL 2) such that
A39: q1 = x and
A40: q1 `2 <= sn * |.q1.| and
q1 `1 <= 0 ;
x in K1 by A38, XBOOLE_0:def 4;
then A41: ex q2 being Point of (TOP-REAL 2) st
( q2 = x & q2 `1 <= 0 & q2 <> 0. (TOP-REAL 2) ) by A3;
(q1 `2 ) / |.q1.| <= (sn * |.q1.|) / |.q1.| by A40, XREAL_1:74;
then (q1 `2 ) / |.q1.| <= sn by A39, A41, TOPRNS_1:25, XCMPLX_1:90;
hence x in K11 by A39, A41; :: thesis: verum
end;
A42: K004 is closed by Th33;
the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
then ( ((TOP-REAL 2) | K1) | K00 = (TOP-REAL 2) | K001 & f1 = f3 ) by A3, FUNCT_1:82, GOBOARD9:4;
then A43: f1 is continuous by A3, A10, Th30, PRE_TOPC:56;
A44: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def 10;
|[(- (sqrt (1 - (sn ^2 )))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } by A1, A6, A5, A7;
then reconsider K111 = { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A19;
A45: rng ((sn -FanMorphW ) | K111) c= K1
proof
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ((sn -FanMorphW ) | K111) or y in K1 )
assume y in rng ((sn -FanMorphW ) | K111) ; :: thesis: y in K1
then consider x being set such that
A46: x in dom ((sn -FanMorphW ) | K111) and
A47: y = ((sn -FanMorphW ) | K111) . x by FUNCT_1:def 5;
reconsider q = x as Point of (TOP-REAL 2) by A46;
A48: y = (sn -FanMorphW ) . q by A46, A47, FUNCT_1:70;
dom ((sn -FanMorphW ) | K111) = (dom (sn -FanMorphW )) /\ K111 by RELAT_1:90
.= the carrier of (TOP-REAL 2) /\ K111 by FUNCT_2:def 1
.= K111 by XBOOLE_1:28 ;
then A49: ex p2 being Point of (TOP-REAL 2) st
( p2 = q & (p2 `2 ) / |.p2.| <= sn & p2 `1 <= 0 & p2 <> 0. (TOP-REAL 2) ) by A46;
then A50: ((q `2 ) / |.q.|) - sn <= 0 by XREAL_1:49;
|.q.| <> 0 by A49, TOPRNS_1:25;
then A51: |.q.| ^2 > 0 ^2 by SQUARE_1:74;
set q4 = |[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|;
A52: |[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| `2 = |.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)) by EUCLID:56;
A53: 1 + sn > 0 by A3, XREAL_1:150;
0 <= (q `1 ) ^2 by XREAL_1:65;
then ( |.q.| ^2 = ((q `1 ) ^2 ) + ((q `2 ) ^2 ) & 0 + ((q `2 ) ^2 ) <= ((q `1 ) ^2 ) + ((q `2 ) ^2 ) ) by JGRAPH_3:10, XREAL_1:9;
then ((q `2 ) ^2 ) / (|.q.| ^2 ) <= (|.q.| ^2 ) / (|.q.| ^2 ) by XREAL_1:74;
then ((q `2 ) ^2 ) / (|.q.| ^2 ) <= 1 by A51, XCMPLX_1:60;
then ((q `2 ) / |.q.|) ^2 <= 1 by XCMPLX_1:77;
then - 1 <= (q `2 ) / |.q.| by SQUARE_1:121;
then (- 1) - sn <= ((q `2 ) / |.q.|) - sn by XREAL_1:11;
then (- (1 + sn)) / (1 + sn) <= (((q `2 ) / |.q.|) - sn) / (1 + sn) by A53, XREAL_1:74;
then - 1 <= (((q `2 ) / |.q.|) - sn) / (1 + sn) by A53, XCMPLX_1:198;
then A54: ((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 <= 1 ^2 by A53, A50, SQUARE_1:119;
then A55: 1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ) >= 0 by XREAL_1:50;
1 - ((- ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 ) >= 0 by A54, XREAL_1:50;
then 1 - (((- (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 ) >= 0 by XCMPLX_1:188;
then sqrt (1 - (((- (((q `2 ) / |.q.|) - sn)) / (1 + sn)) ^2 )) >= 0 by SQUARE_1:def 4;
then sqrt (1 - (((- (((q `2 ) / |.q.|) - sn)) ^2 ) / ((1 + sn) ^2 ))) >= 0 by XCMPLX_1:77;
then sqrt (1 - (((((q `2 ) / |.q.|) - sn) ^2 ) / ((1 + sn) ^2 ))) >= 0 ;
then A56: sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )) >= 0 by XCMPLX_1:77;
A57: |[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| `1 = |.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))) by EUCLID:56;
then A58: (|[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| `1 ) ^2 = (|.q.| ^2 ) * ((sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))) ^2 )
.= (|.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )) by A55, SQUARE_1:def 4 ;
|.|[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|.| ^2 = ((|[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| `1 ) ^2 ) + ((|[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| `2 ) ^2 ) by JGRAPH_3:10
.= |.q.| ^2 by A52, A58 ;
then A59: |[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| <> 0. (TOP-REAL 2) by A51, TOPRNS_1:24;
(sn -FanMorphW ) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A3, A49, Th25;
hence y in K1 by A3, A48, A57, A56, A59; :: thesis: verum
end;
dom ((sn -FanMorphW ) | K111) = K111 by A35, RELAT_1:91
.= the carrier of ((TOP-REAL 2) | K111) by PRE_TOPC:29 ;
then reconsider f4 = (sn -FanMorphW ) | K111 as Function of ((TOP-REAL 2) | K111),((TOP-REAL 2) | K1) by A16, A45, FUNCT_2:4;
the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
then ( ((TOP-REAL 2) | K1) | K11 = (TOP-REAL 2) | K111 & f2 = f4 ) by A3, FUNCT_1:82, GOBOARD9:4;
then A60: f2 is continuous by A3, A10, Th31, PRE_TOPC:56;
set T1 = ((TOP-REAL 2) | K1) | K00;
set T2 = ((TOP-REAL 2) | K1) | K11;
A61: [#] (((TOP-REAL 2) | K1) | K11) = K11 by PRE_TOPC:def 10;
K11 c= K004 /\ K1
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K11 or x in K004 /\ K1 )
assume x in K11 ; :: thesis: x in K004 /\ K1
then consider p being Point of (TOP-REAL 2) such that
A62: p = x and
A63: (p `2 ) / |.p.| <= sn and
A64: p `1 <= 0 and
A65: p <> 0. (TOP-REAL 2) ;
((p `2 ) / |.p.|) * |.p.| <= sn * |.p.| by A63, XREAL_1:66;
then p `2 <= sn * |.p.| by A65, TOPRNS_1:25, XCMPLX_1:88;
then A66: x in K004 by A62, A64;
x in K1 by A3, A62, A64, A65;
hence x in K004 /\ K1 by A66, XBOOLE_0:def 4; :: thesis: verum
end;
then K11 = K004 /\ ([#] ((TOP-REAL 2) | K1)) by A44, A37, XBOOLE_0:def 10;
then A67: K11 is closed by A42, PRE_TOPC:43;
A68: K003 /\ K1 c= K00
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K003 /\ K1 or x in K00 )
assume A69: x in K003 /\ K1 ; :: thesis: x in K00
then x in K003 by XBOOLE_0:def 4;
then consider q1 being Point of (TOP-REAL 2) such that
A70: q1 = x and
A71: q1 `2 >= sn * |.q1.| and
q1 `1 <= 0 ;
x in K1 by A69, XBOOLE_0:def 4;
then A72: ex q2 being Point of (TOP-REAL 2) st
( q2 = x & q2 `1 <= 0 & q2 <> 0. (TOP-REAL 2) ) by A3;
(q1 `2 ) / |.q1.| >= (sn * |.q1.|) / |.q1.| by A71, XREAL_1:74;
then (q1 `2 ) / |.q1.| >= sn by A70, A72, TOPRNS_1:25, XCMPLX_1:90;
hence x in K00 by A70, A72; :: thesis: verum
end;
A73: the carrier of ((TOP-REAL 2) | K1) = K0 by PRE_TOPC:29;
A74: D <> {} ;
A75: [#] (((TOP-REAL 2) | K1) | K00) = K00 by PRE_TOPC:def 10;
A76: for p being set st p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) holds
f1 . p = f2 . p
proof
let p be set ; :: thesis: ( p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) implies f1 . p = f2 . p )
assume A77: p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) ; :: thesis: f1 . p = f2 . p
then p in K00 by A75, XBOOLE_0:def 4;
hence f1 . p = f . p by FUNCT_1:72
.= f2 . p by A61, A77, FUNCT_1:72 ;
:: thesis: verum
end;
K00 c= K003 /\ K1
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K00 or x in K003 /\ K1 )
assume x in K00 ; :: thesis: x in K003 /\ K1
then consider p being Point of (TOP-REAL 2) such that
A78: p = x and
A79: (p `2 ) / |.p.| >= sn and
A80: p `1 <= 0 and
A81: p <> 0. (TOP-REAL 2) ;
((p `2 ) / |.p.|) * |.p.| >= sn * |.p.| by A79, XREAL_1:66;
then p `2 >= sn * |.p.| by A81, TOPRNS_1:25, XCMPLX_1:88;
then A82: x in K003 by A78, A80;
x in K1 by A3, A78, A80, A81;
hence x in K003 /\ K1 by A82, XBOOLE_0:def 4; :: thesis: verum
end;
then K00 = K003 /\ ([#] ((TOP-REAL 2) | K1)) by A44, A68, XBOOLE_0:def 10;
then A83: K00 is closed by A36, PRE_TOPC:43;
A84: K1 c= K00 \/ K11
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K1 or x in K00 \/ K11 )
assume x in K1 ; :: thesis: x in K00 \/ K11
then consider p being Point of (TOP-REAL 2) such that
A85: ( p = x & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) by A3;
per cases ( (p `2 ) / |.p.| >= sn or (p `2 ) / |.p.| < sn ) ;
suppose (p `2 ) / |.p.| >= sn ; :: thesis: x in K00 \/ K11
then x in K00 by A85;
hence x in K00 \/ K11 by XBOOLE_0:def 3; :: thesis: verum
end;
suppose (p `2 ) / |.p.| < sn ; :: thesis: x in K00 \/ K11
then x in K11 by A85;
hence x in K00 \/ K11 by XBOOLE_0:def 3; :: thesis: verum
end;
end;
end;
then ([#] (((TOP-REAL 2) | K1) | K00)) \/ ([#] (((TOP-REAL 2) | K1) | K11)) = [#] ((TOP-REAL 2) | K1) by A75, A61, A44, XBOOLE_0:def 10;
then consider h being Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) such that
A86: h = f1 +* f2 and
A87: h is continuous by A75, A61, A83, A67, A43, A60, A76, JGRAPH_2:9;
A88: dom h = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def 1;
A89: dom f1 = the carrier of (((TOP-REAL 2) | K1) | K00) by FUNCT_2:def 1
.= K00 by PRE_TOPC:29 ;
A90: for y being set st y in dom h holds
h . y = f . y
proof
let y be set ; :: thesis: ( y in dom h implies h . y = f . y )
assume A91: y in dom h ; :: thesis: h . y = f . y
now
per cases ( ( y in K00 & not y in K11 ) or y in K11 ) by A84, A88, A73, A91, XBOOLE_0:def 3;
case A92: ( y in K00 & not y in K11 ) ; :: thesis: h . y = f . y
then y in (dom f1) \/ (dom f2) by A89, XBOOLE_0:def 3;
hence h . y = f1 . y by A17, A86, A92, FUNCT_4:def 1
.= f . y by A92, FUNCT_1:72 ;
:: thesis: verum
end;
case A93: y in K11 ; :: thesis: h . y = f . y
then y in (dom f1) \/ (dom f2) by A17, XBOOLE_0:def 3;
hence h . y = f2 . y by A17, A86, A93, FUNCT_4:def 1
.= f . y by A93, FUNCT_1:72 ;
:: thesis: verum
end;
end;
end;
hence h . y = f . y ; :: thesis: verum
end;
K0 = the carrier of ((TOP-REAL 2) | K0) by PRE_TOPC:29
.= dom f by A74, FUNCT_2:def 1 ;
hence f is continuous by A87, A88, A90, FUNCT_1:9, PRE_TOPC:29; :: thesis: verum