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 -FanMorphE ) | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0.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 -FanMorphE ) | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0.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 -FanMorphE ) | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0.REAL 2 ) } implies f is continuous )
assume A1: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE ) | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0.REAL 2 ) } ) ; :: thesis: f is continuous
set cn = sqrt (1 - (sn ^2 ));
set p0 = |[(sqrt (1 - (sn ^2 ))),sn]|;
A2: |[(sqrt (1 - (sn ^2 ))),sn]| `2 = sn by EUCLID:56;
A3: |[(sqrt (1 - (sn ^2 ))),sn]| `1 = sqrt (1 - (sn ^2 )) by EUCLID:56;
sn ^2 < 1 ^2 by A1, SQUARE_1:120;
then A4: 1 - (sn ^2 ) > 0 by XREAL_1:52;
then A5: |[(sqrt (1 - (sn ^2 ))),sn]| <> 0.REAL 2 by A3, JGRAPH_2:11, SQUARE_1:93;
A6: - (- (sqrt (1 - (sn ^2 )))) > 0 by A4, SQUARE_1:93;
A7: |[(sqrt (1 - (sn ^2 ))),sn]| `1 > 0 by A3, A4, SQUARE_1:93;
A8: |.|[(sqrt (1 - (sn ^2 ))),sn]|.| = sqrt (((sqrt (1 - (sn ^2 ))) ^2 ) + (sn ^2 )) by A2, A3, JGRAPH_3:10;
(sqrt (1 - (sn ^2 ))) ^2 = 1 - (sn ^2 ) by A4, SQUARE_1:def 4;
then A9: (|[(sqrt (1 - (sn ^2 ))),sn]| `2 ) / |.|[(sqrt (1 - (sn ^2 ))),sn]|.| = sn by A8, EUCLID:56, SQUARE_1:83;
|[(sqrt (1 - (sn ^2 ))),sn]| in K0 by A1, A7, JGRAPH_2:11;
then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ;
( |[(sqrt (1 - (sn ^2 ))),sn]| in the carrier of (TOP-REAL 2) & not |[(sqrt (1 - (sn ^2 ))),sn]| in {(0.REAL 2)} ) by A5, TARSKI:def 1;
then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A1, XBOOLE_0:def 5;
A10: |[(sqrt (1 - (sn ^2 ))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| >= sn & p `1 >= 0 & p <> 0.REAL 2 ) } by A7, A9, JGRAPH_2:11;
defpred S1[ Point of (TOP-REAL 2)] means ( ($1 `2 ) / |.$1.| >= sn & $1 `1 >= 0 & $1 <> 0.REAL 2 );
A11: { p where p is Point of (TOP-REAL 2) : S1[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch 7();
 A12: { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| >= sn & p `1 >= 0 & p <> 0.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.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.REAL 2 ) } ; :: thesis: x in K1
then consider p being Point of (TOP-REAL 2) such that
A13: ( p = x & (p `2 ) / |.p.| >= sn & p `1 >= 0 & p <> 0.REAL 2 ) ;
thus x in K1 by A1, A13; :: thesis: verum
end;
then reconsider K00 = { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| >= sn & p `1 >= 0 & p <> 0.REAL 2 ) } as non empty Subset of ((TOP-REAL 2) | K1) by A10, PRE_TOPC:29;
reconsider K001 = { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| >= sn & p `1 >= 0 & p <> 0.REAL 2 ) } as non empty Subset of (TOP-REAL 2) by A10, A11;
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) ;
A14: |[(sqrt (1 - (sn ^2 ))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 >= 0 & p <> 0.REAL 2 ) } by A3, A6, A9, JGRAPH_2:11;
A15: { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 >= 0 & p <> 0.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.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.REAL 2 ) } ; :: thesis: x in K1
then consider p being Point of (TOP-REAL 2) such that
A16: ( p = x & (p `2 ) / |.p.| <= sn & p `1 >= 0 & p <> 0.REAL 2 ) ;
thus x in K1 by A1, A16; :: thesis: verum
end;
then reconsider K11 = { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 >= 0 & p <> 0.REAL 2 ) } as non empty Subset of ((TOP-REAL 2) | K1) by A14, PRE_TOPC:29;
A17: |[(sqrt (1 - (sn ^2 ))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 >= 0 & p <> 0.REAL 2 ) } by A3, A6, A9, JGRAPH_2:11;
defpred S3[ Point of (TOP-REAL 2)] means ( ($1 `2 ) / |.$1.| <= sn & $1 `1 >= 0 & $1 <> 0.REAL 2 );
{ p where p is Point of (TOP-REAL 2) : S3[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch 7();
 then reconsider K111 = { p where p is Point of (TOP-REAL 2) : ( (p `2 ) / |.p.| <= sn & p `1 >= 0 & p <> 0.REAL 2 ) } as non empty Subset of (TOP-REAL 2) by A17;
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) ;
the carrier of ((TOP-REAL 2) | B0) = the carrier of ((TOP-REAL 2) | D) ;
then A18: dom f = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def 1
.= K1 by PRE_TOPC:29 ;
then A19: dom (f | K00) = K00 by A12, RELAT_1:91
.= the carrier of (((TOP-REAL 2) | K1) | K00) by PRE_TOPC:29 ;
the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:29;
then rng (f | K00) c= D ;
then reconsider f1 = f | K00 as Function of (((TOP-REAL 2) | K1) | K00),((TOP-REAL 2) | D) by A19, FUNCT_2:4;
A20: dom f1 = the carrier of (((TOP-REAL 2) | K1) | K00) by FUNCT_2:def 1
.= K00 by PRE_TOPC:29 ;
A21: dom (sn -FanMorphE ) = the carrier of (TOP-REAL 2) by FUNCT_2:def 1;
then A22: dom ((sn -FanMorphE ) | K001) = K001 by RELAT_1:91
.= the carrier of ((TOP-REAL 2) | K001) by PRE_TOPC:29 ;
A23: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
rng ((sn -FanMorphE ) | K001) c= K1
proof
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ((sn -FanMorphE ) | K001) or y in K1 )
assume y in rng ((sn -FanMorphE ) | K001) ; :: thesis: y in K1
then consider x being set such that
A24: ( x in dom ((sn -FanMorphE ) | K001) & y = ((sn -FanMorphE ) | K001) . x ) by FUNCT_1:def 5;
A25: dom ((sn -FanMorphE ) | K001) = (dom (sn -FanMorphE )) /\ K001 by FUNCT_1:68
.= the carrier of (TOP-REAL 2) /\ K001 by FUNCT_2:def 1
.= K001 by XBOOLE_1:28 ;
reconsider q = x as Point of (TOP-REAL 2) by A24;
A26: y = (sn -FanMorphE ) . q by A24, FUNCT_1:70;
consider p2 being Point of (TOP-REAL 2) such that
A27: ( p2 = q & (p2 `2 ) / |.p2.| >= sn & p2 `1 >= 0 & p2 <> 0.REAL 2 ) by A24, A25;
A28: (sn -FanMorphE ) . q = |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A1, A27, Th91;
set q4 = |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|;
|.q.| <> 0 by A27, TOPRNS_1:25;
then A29: |.q.| ^2 > 0 ^2 by SQUARE_1:74;
A30: 1 - sn > 0 by A1, XREAL_1:151;
A31: ( |[(|.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 ))) & |[(|.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;
A32: ((q `2 ) / |.q.|) - sn >= 0 by A27, XREAL_1:50;
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 A29, XCMPLX_1:60;
then ((q `2 ) / |.q.|) ^2 <= 1 by XCMPLX_1:77;
then 1 >= (q `2 ) / |.q.| by SQUARE_1:121;
then A33: 1 - sn >= ((q `2 ) / |.q.|) - sn by XREAL_1:11;
A34: - (1 - sn) <= - 0 by A30;
- (1 - sn) <= - (((q `2 ) / |.q.|) - sn) by A33, XREAL_1:26;
then (- (1 - sn)) / (1 - sn) <= (- (((q `2 ) / |.q.|) - sn)) / (1 - sn) by A30, XREAL_1:74;
then A35: - 1 <= (- (((q `2 ) / |.q.|) - sn)) / (1 - sn) by A30, XCMPLX_1:198;
- (- (1 - sn)) >= - (((q `2 ) / |.q.|) - sn) by A32, A34, XREAL_1:26;
then (- (((q `2 ) / |.q.|) - sn)) / (1 - sn) <= 1 by A30, XREAL_1:187;
then ((- (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A35, SQUARE_1:119;
then A36: 1 - (((- (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 ) >= 0 by XREAL_1:50;
then A37: 1 - ((- ((((q `2 ) / |.q.|) - sn) / (1 - sn))) ^2 ) >= 0 by XCMPLX_1:188;
sqrt (1 - (((- (((q `2 ) / |.q.|) - sn)) / (1 - sn)) ^2 )) >= 0 by A36, 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 A38: sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )) >= 0 by XCMPLX_1:77;
A39: (|[(|.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 ) by A31
.= (|.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )) by A37, 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 A31, A39 ;
then ( |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| `1 >= 0 ^2 & |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| <> 0.REAL 2 ) by A29, A31, A38, TOPRNS_1:24;
hence y in K1 by A1, A26, A28; :: thesis: verum
end;
then reconsider f3 = (sn -FanMorphE ) | K001 as Function of ((TOP-REAL 2) | K001),((TOP-REAL 2) | K1) by A22, A23, FUNCT_2:4;
A40: dom (f | K11) = K11 by A15, A18, RELAT_1:91
.= the carrier of (((TOP-REAL 2) | K1) | K11) by PRE_TOPC:29 ;
the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:29;
then rng (f | K11) c= D ;
then reconsider f2 = f | K11 as Function of (((TOP-REAL 2) | K1) | K11),((TOP-REAL 2) | D) by A40, FUNCT_2:4;
A41: dom f2 = the carrier of (((TOP-REAL 2) | K1) | K11) by FUNCT_2:def 1
.= K11 by PRE_TOPC:29 ;
A42: dom ((sn -FanMorphE ) | K111) = K111 by A21, RELAT_1:91
.= the carrier of ((TOP-REAL 2) | K111) by PRE_TOPC:29 ;
A43: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
rng ((sn -FanMorphE ) | K111) c= K1
proof
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ((sn -FanMorphE ) | K111) or y in K1 )
assume y in rng ((sn -FanMorphE ) | K111) ; :: thesis: y in K1
then consider x being set such that
A44: ( x in dom ((sn -FanMorphE ) | K111) & y = ((sn -FanMorphE ) | K111) . x ) by FUNCT_1:def 5;
A45: dom ((sn -FanMorphE ) | K111) = (dom (sn -FanMorphE )) /\ K111 by FUNCT_1:68
.= the carrier of (TOP-REAL 2) /\ K111 by FUNCT_2:def 1
.= K111 by XBOOLE_1:28 ;
reconsider q = x as Point of (TOP-REAL 2) by A44;
A46: y = (sn -FanMorphE ) . q by A44, FUNCT_1:70;
consider p2 being Point of (TOP-REAL 2) such that
A47: ( p2 = q & (p2 `2 ) / |.p2.| <= sn & p2 `1 >= 0 & p2 <> 0.REAL 2 ) by A44, A45;
A48: (sn -FanMorphE ) . q = |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| by A1, A47, Th91;
set q4 = |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]|;
|.q.| <> 0 by A47, TOPRNS_1:25;
then A49: |.q.| ^2 > 0 ^2 by SQUARE_1:74;
A50: 1 + sn > 0 by A1, XREAL_1:150;
A51: ( |[(|.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 ))) & |[(|.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;
A52: ((q `2 ) / |.q.|) - sn <= 0 by A47, XREAL_1:49;
A53: |.q.| ^2 = ((q `1 ) ^2 ) + ((q `2 ) ^2 ) by JGRAPH_3:10;
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 ) <= (|.q.| ^2 ) / (|.q.| ^2 ) by A53, XREAL_1:74;
then ((q `2 ) ^2 ) / (|.q.| ^2 ) <= 1 by A49, XCMPLX_1:60;
then ((q `2 ) / |.q.|) ^2 <= 1 by XCMPLX_1:77;
then - 1 <= (q `2 ) / |.q.| by SQUARE_1:121;
then A54: (- 1) - sn <= ((q `2 ) / |.q.|) - sn by XREAL_1:11;
(1 + sn) / (1 + sn) >= (((q `2 ) / |.q.|) - sn) / (1 + sn) by A50, A52, XREAL_1:74;
then A55: 1 >= (((q `2 ) / |.q.|) - sn) / (1 + sn) by A50, XCMPLX_1:60;
(- (1 + sn)) / (1 + sn) <= (((q `2 ) / |.q.|) - sn) / (1 + sn) by A50, A54, XREAL_1:74;
then - 1 <= (((q `2 ) / |.q.|) - sn) / (1 + sn) by A50, XCMPLX_1:198;
then A56: ((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 <= 1 ^2 by A55, SQUARE_1:119;
then A57: 1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 ) >= 0 by XREAL_1:50;
1 - ((- ((((q `2 ) / |.q.|) - sn) / (1 + sn))) ^2 ) >= 0 by A56, 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 sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )) >= 0 by XCMPLX_1:77;
then A58: |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| `1 >= 0 by A51;
A59: (|[(|.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 ) by A51
.= (|.q.| ^2 ) * (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )) by A57, 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 A51, A59 ;
then |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 + sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 + sn)))]| <> 0.REAL 2 by A49, TOPRNS_1:24;
hence y in K1 by A1, A46, A48, A58; :: thesis: verum
end;
then reconsider f4 = (sn -FanMorphE ) | K111 as Function of ((TOP-REAL 2) | K111),((TOP-REAL 2) | K1) by A42, A43, FUNCT_2:4;
set T1 = ((TOP-REAL 2) | K1) | K00;
set T2 = ((TOP-REAL 2) | K1) | K11;
A60: [#] (((TOP-REAL 2) | K1) | K00) = K00 by PRE_TOPC:def 10;
A61: [#] (((TOP-REAL 2) | K1) | K11) = K11 by PRE_TOPC:def 10;
A62: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def 10;
A63: 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
A64: ( p = x & p `1 >= 0 & p <> 0.REAL 2 ) by A1;
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 A64;
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 A64;
hence x in K00 \/ K11 by XBOOLE_0:def 3; :: thesis: verum
end;
end;
end;
then A65: ([#] (((TOP-REAL 2) | K1) | K00)) \/ ([#] (((TOP-REAL 2) | K1) | K11)) = [#] ((TOP-REAL 2) | K1) by A60, A61, A62, XBOOLE_0:def 10;
A66: K003 is closed by Th98;
A67: 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 x in K003 /\ K1 ; :: thesis: x in K00
then A68: ( x in K003 & x in K1 ) by XBOOLE_0:def 4;
then consider q1 being Point of (TOP-REAL 2) such that
A69: ( q1 = x & q1 `2 >= sn * |.q1.| & q1 `1 >= 0 ) ;
consider q2 being Point of (TOP-REAL 2) such that
A70: ( q2 = x & q2 `1 >= 0 & q2 <> 0.REAL 2 ) by A1, A68;
(q1 `2 ) / |.q1.| >= (sn * |.q1.|) / |.q1.| by A69, XREAL_1:74;
then (q1 `2 ) / |.q1.| >= sn by A69, A70, TOPRNS_1:25, XCMPLX_1:90;
hence x in K00 by A69, A70; :: 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
A72: ( p = x & (p `2 ) / |.p.| >= sn & p `1 >= 0 & p <> 0.REAL 2 ) ;
((p `2 ) / |.p.|) * |.p.| >= sn * |.p.| by A72, XREAL_1:66;
then p `2 >= sn * |.p.| by A72, TOPRNS_1:25, XCMPLX_1:88;
then A73: x in K003 by A72;
x in K1 by A1, A72;
hence x in K003 /\ K1 by A73, XBOOLE_0:def 4; :: thesis: verum
end;
then K00 = K003 /\ ([#] ((TOP-REAL 2) | K1)) by A62, A67, XBOOLE_0:def 10;
then A74: K00 is closed by A66, PRE_TOPC:43;
A75: K004 is closed by Th99;
A76: 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 x in K004 /\ K1 ; :: thesis: x in K11
then A77: ( x in K004 & x in K1 ) by XBOOLE_0:def 4;
then consider q1 being Point of (TOP-REAL 2) such that
A78: ( q1 = x & q1 `2 <= sn * |.q1.| & q1 `1 >= 0 ) ;
consider q2 being Point of (TOP-REAL 2) such that
A79: ( q2 = x & q2 `1 >= 0 & q2 <> 0.REAL 2 ) by A1, A77;
(q1 `2 ) / |.q1.| <= (sn * |.q1.|) / |.q1.| by A78, XREAL_1:74;
then (q1 `2 ) / |.q1.| <= sn by A78, A79, TOPRNS_1:25, XCMPLX_1:90;
hence x in K11 by A78, A79; :: thesis: verum
end;
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
A81: ( p = x & (p `2 ) / |.p.| <= sn & p `1 >= 0 & p <> 0.REAL 2 ) ;
((p `2 ) / |.p.|) * |.p.| <= sn * |.p.| by A81, XREAL_1:66;
then p `2 <= sn * |.p.| by A81, TOPRNS_1:25, XCMPLX_1:88;
then A82: x in K004 by A81;
x in K1 by A1, A81;
hence x in K004 /\ K1 by A82, XBOOLE_0:def 4; :: thesis: verum
end;
then K11 = K004 /\ ([#] ((TOP-REAL 2) | K1)) by A62, A76, XBOOLE_0:def 10;
then A83: K11 is closed by A75, PRE_TOPC:43;
A84: ((TOP-REAL 2) | K1) | K00 = (TOP-REAL 2) | K001 by GOBOARD9:4;
K1 c= D
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K1 or x in D )
assume x in K1 ; :: thesis: x in D
then consider p6 being Point of (TOP-REAL 2) such that
A85: ( p6 = x & p6 `1 >= 0 & p6 <> 0.REAL 2 ) by A1;
( x in the carrier of (TOP-REAL 2) & not x in {(0.REAL 2)} ) by A85, TARSKI:def 1;
hence x in D by A1, XBOOLE_0:def 5; :: thesis: verum
end;
then D = K1 \/ D by XBOOLE_1:12;
then A86: (TOP-REAL 2) | K1 is SubSpace of (TOP-REAL 2) | D by TOPMETR:5;
the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
then f1 = f3 by A1, FUNCT_1:82;
then A87: f1 is continuous by A1, A84, A86, Th96, PRE_TOPC:56;
A88: ((TOP-REAL 2) | K1) | K11 = (TOP-REAL 2) | K111 by GOBOARD9:4;
the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
then f2 = f4 by A1, FUNCT_1:82;
then A89: f2 is continuous by A1, A86, A88, Th97, PRE_TOPC:56;
A90: D <> {} ;
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 A92: p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) ; :: thesis: f1 . p = f2 . p
then ( p in K00 & p in K11 ) by A60, A61, XBOOLE_0:def 4;
hence f1 . p = f . p by FUNCT_1:72
.= f2 . p by A61, A92, FUNCT_1:72 ;
:: thesis: verum
end;
then consider h being Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) such that
A93: ( h = f1 +* f2 & h is continuous ) by A60, A61, A65, A74, A83, A87, A89, JGRAPH_2:9;
A94: dom h = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def 1;
A95: the carrier of ((TOP-REAL 2) | K1) = K0 by PRE_TOPC:29;
A96: K0 = the carrier of ((TOP-REAL 2) | K0) by PRE_TOPC:29
.= dom f by A90, FUNCT_2:def 1 ;
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 A97: y in dom h ; :: thesis: h . y = f . y
now
per cases ( ( y in K00 & not y in K11 ) or y in K11 ) by A63, A94, A95, A97, XBOOLE_0:def 3;
suppose A98: ( y in K00 & not y in K11 ) ; :: thesis: h . y = f . y
then y in (dom f1) \/ (dom f2) by A20, XBOOLE_0:def 3;
hence h . y = f1 . y by A41, A93, A98, FUNCT_4:def 1
.= f . y by A98, FUNCT_1:72 ;
:: thesis: verum
end;
suppose A99: y in K11 ; :: thesis: h . y = f . y
then y in (dom f1) \/ (dom f2) by A41, XBOOLE_0:def 3;
hence h . y = f2 . y by A41, A93, A99, FUNCT_4:def 1
.= f . y by A99, FUNCT_1:72 ;
:: thesis: verum
end;
end;
end;
hence h . y = f . y ; :: thesis: verum
end;
hence f is continuous by A93, A94, A95, A96, FUNCT_1:9; :: thesis: verum