let cn be Real; for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
(cn -FanMorphN) . x in K0
let x, K0 be set ; ( - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } implies (cn -FanMorphN) . x in K0 )
assume A1:
( - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } )
; (cn -FanMorphN) . x in K0
then consider p being Point of (TOP-REAL 2) such that
A2:
p = x
and
A3:
p `2 >= 0
and
A4:
p <> 0. (TOP-REAL 2)
;
then A6:
|.p.| ^2 > 0
by SQUARE_1:12;
per cases
( (p `1) / |.p.| <= cn or (p `1) / |.p.| > cn )
;
suppose A7:
(p `1) / |.p.| <= cn
;
(cn -FanMorphN) . x in K0reconsider p9 =
(cn -FanMorphN) . p as
Point of
(TOP-REAL 2) ;
(cn -FanMorphN) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))))]|
by A1, A3, A4, A7, Th51;
then A8:
p9 `2 = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))
by EUCLID:52;
A9:
|.p.| ^2 = ((p `1) ^2) + ((p `2) ^2)
by JGRAPH_3:1;
A10:
1
+ cn > 0
by A1, XREAL_1:148;
per cases
( p `2 = 0 or p `2 <> 0 )
;
suppose
p `2 <> 0
;
(cn -FanMorphN) . x in K0then
0 + ((p `1) ^2) < ((p `1) ^2) + ((p `2) ^2)
by SQUARE_1:12, XREAL_1:8;
then
((p `1) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2)
by A6, A9, XREAL_1:74;
then
((p `1) ^2) / (|.p.| ^2) < 1
by A6, XCMPLX_1:60;
then
((p `1) / |.p.|) ^2 < 1
by XCMPLX_1:76;
then
- 1
< (p `1) / |.p.|
by SQUARE_1:52;
then
(- 1) - cn < ((p `1) / |.p.|) - cn
by XREAL_1:9;
then
((- 1) * (1 + cn)) / (1 + cn) < (((p `1) / |.p.|) - cn) / (1 + cn)
by A10, XREAL_1:74;
then A11:
- 1
< (((p `1) / |.p.|) - cn) / (1 + cn)
by A10, XCMPLX_1:89;
((p `1) / |.p.|) - cn <= 0
by A7, XREAL_1:47;
then
1
^2 > ((((p `1) / |.p.|) - cn) / (1 + cn)) ^2
by A10, A11, SQUARE_1:50;
then
1
- (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2) > 0
by XREAL_1:50;
then
sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)) > 0
by SQUARE_1:25;
then
|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) > 0
by A5, XREAL_1:129;
hence
(cn -FanMorphN) . x in K0
by A1, A2, A8, JGRAPH_2:3;
verum end; end; end; suppose A12:
(p `1) / |.p.| > cn
;
(cn -FanMorphN) . x in K0reconsider p9 =
(cn -FanMorphN) . p as
Point of
(TOP-REAL 2) ;
(cn -FanMorphN) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))))]|
by A1, A3, A4, A12, Th51;
then A13:
p9 `2 = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))
by EUCLID:52;
A14:
|.p.| ^2 = ((p `1) ^2) + ((p `2) ^2)
by JGRAPH_3:1;
A15:
1
- cn > 0
by A1, XREAL_1:149;
per cases
( p `2 = 0 or p `2 <> 0 )
;
suppose
p `2 <> 0
;
(cn -FanMorphN) . x in K0then
0 + ((p `1) ^2) < ((p `1) ^2) + ((p `2) ^2)
by SQUARE_1:12, XREAL_1:8;
then
((p `1) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2)
by A6, A14, XREAL_1:74;
then
((p `1) ^2) / (|.p.| ^2) < 1
by A6, XCMPLX_1:60;
then
((p `1) / |.p.|) ^2 < 1
by XCMPLX_1:76;
then
(p `1) / |.p.| < 1
by SQUARE_1:52;
then
((p `1) / |.p.|) - cn < 1
- cn
by XREAL_1:9;
then
(((p `1) / |.p.|) - cn) / (1 - cn) < (1 - cn) / (1 - cn)
by A15, XREAL_1:74;
then A16:
(((p `1) / |.p.|) - cn) / (1 - cn) < 1
by A15, XCMPLX_1:60;
(
- (1 - cn) < - 0 &
((p `1) / |.p.|) - cn >= cn - cn )
by A12, A15, XREAL_1:9, XREAL_1:24;
then
((- 1) * (1 - cn)) / (1 - cn) < (((p `1) / |.p.|) - cn) / (1 - cn)
by A15, XREAL_1:74;
then
- 1
< (((p `1) / |.p.|) - cn) / (1 - cn)
by A15, XCMPLX_1:89;
then
1
^2 > ((((p `1) / |.p.|) - cn) / (1 - cn)) ^2
by A16, SQUARE_1:50;
then
1
- (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2) > 0
by XREAL_1:50;
then
sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)) > 0
by SQUARE_1:25;
then
p9 `2 > 0
by A5, A13, XREAL_1:129;
hence
(cn -FanMorphN) . x in K0
by A1, A2, JGRAPH_2:3;
verum end; end; end; end;