let sn be Real; :: thesis: ( - 1 < sn & sn < 1 implies ( sn -FanMorphE is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (sn -FanMorphE ) = the carrier of (TOP-REAL 2) ) )
assume A1:
( - 1 < sn & sn < 1 )
; :: thesis: ( sn -FanMorphE is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (sn -FanMorphE ) = the carrier of (TOP-REAL 2) )
thus
sn -FanMorphE is Function of (TOP-REAL 2),(TOP-REAL 2)
; :: thesis: rng (sn -FanMorphE ) = the carrier of (TOP-REAL 2)
for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = sn -FanMorphE holds
rng (sn -FanMorphE ) = the carrier of (TOP-REAL 2)
proof
let f be
Function of
(TOP-REAL 2),
(TOP-REAL 2);
:: thesis: ( f = sn -FanMorphE implies rng (sn -FanMorphE ) = the carrier of (TOP-REAL 2) )
assume A2:
f = sn -FanMorphE
;
:: thesis: rng (sn -FanMorphE ) = the carrier of (TOP-REAL 2)
A3:
dom f = the
carrier of
(TOP-REAL 2)
by FUNCT_2:def 1;
the
carrier of
(TOP-REAL 2) c= rng f
proof
let y be
set ;
:: according to TARSKI:def 3 :: thesis: ( not y in the carrier of (TOP-REAL 2) or y in rng f )
assume
y in the
carrier of
(TOP-REAL 2)
;
:: thesis: y in rng f
then reconsider p2 =
y as
Point of
(TOP-REAL 2) ;
set q =
p2;
now per cases
( p2 `1 <= 0 or ( (p2 `2 ) / |.p2.| >= 0 & p2 `1 >= 0 & p2 <> 0. (TOP-REAL 2) ) or ( (p2 `2 ) / |.p2.| < 0 & p2 `1 >= 0 & p2 <> 0. (TOP-REAL 2) ) )
by JGRAPH_2:11;
suppose A4:
(
(p2 `2 ) / |.p2.| >= 0 &
p2 `1 >= 0 &
p2 <> 0. (TOP-REAL 2) )
;
:: thesis: ex x being set st
( x in dom (sn -FanMorphE ) & y = (sn -FanMorphE ) . x )set px =
|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|;
A5:
(
|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]| `1 = |.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 ))) &
|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]| `2 = |.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) )
by EUCLID:56;
then A6:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 =
((|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))) ^2 ) + ((|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn)) ^2 )
by JGRAPH_3:10
.=
((|.p2.| ^2 ) * ((sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 ))) ^2 )) + ((|.p2.| ^2 ) * (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 ))
;
A7:
|.p2.| <> 0
by A4, TOPRNS_1:25;
1
- sn >= 0
by A1, XREAL_1:151;
then A8:
((p2 `2 ) / |.p2.|) * (1 - sn) >= 0
by A4;
then A9:
(((p2 `2 ) / |.p2.|) * (1 - sn)) + sn >= 0 + sn
by XREAL_1:9;
A10:
|.p2.| ^2 > 0
by A7, SQUARE_1:74;
- (- (1 + sn)) > 0
by A1, XREAL_1:150;
then
- ((- 1) - sn) > 0
;
then
(- 1) - sn <= ((p2 `2 ) / |.p2.|) * (1 - sn)
by A8;
then A11:
((- 1) - sn) + sn <= (((p2 `2 ) / |.p2.|) * (1 - sn)) + sn
by XREAL_1:9;
A12:
1
- sn > 0
by A1, XREAL_1:151;
A13:
|.p2.| ^2 = ((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )
by JGRAPH_3:10;
0 <= (p2 `1 ) ^2
by XREAL_1:65;
then
0 + ((p2 `2 ) ^2 ) <= ((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )
by XREAL_1:9;
then
((p2 `2 ) ^2 ) / (|.p2.| ^2 ) <= (|.p2.| ^2 ) / (|.p2.| ^2 )
by A13, XREAL_1:74;
then
((p2 `2 ) ^2 ) / (|.p2.| ^2 ) <= 1
by A10, XCMPLX_1:60;
then
((p2 `2 ) / |.p2.|) ^2 <= 1
by XCMPLX_1:77;
then
(p2 `2 ) / |.p2.| <= 1
by SQUARE_1:121;
then
((p2 `2 ) / |.p2.|) * (1 - sn) <= 1
* (1 - sn)
by A12, XREAL_1:66;
then
((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) - sn <= 1
- sn
;
then
(((p2 `2 ) / |.p2.|) * (1 - sn)) + sn <= 1
by XREAL_1:11;
then
1
^2 >= ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2
by A11, SQUARE_1:119;
then A14:
1
- (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 ) >= 0
by XREAL_1:50;
then A15:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 =
((|.p2.| ^2 ) * (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 ))) + ((|.p2.| ^2 ) * (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 ))
by A6, SQUARE_1:def 4
.=
|.p2.| ^2
;
then A16:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| =
sqrt (|.p2.| ^2 )
by SQUARE_1:89
.=
|.p2.|
by SQUARE_1:89
;
sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )) >= 0
by A14, SQUARE_1:def 4;
then
(
(|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]| `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| >= sn &
|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]| `1 >= 0 &
|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]| <> 0. (TOP-REAL 2) )
by A5, A7, A9, A16, TOPRNS_1:24, XCMPLX_1:90;
then A17:
(sn -FanMorphE ) . |[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]| = |[(|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (1 - (((((|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]| `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)) ^2 )))),(|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| * ((((|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]| `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)))]|
by A1, Th91;
A18:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| * ((((|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]| `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)) =
|.p2.| * ((((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) - sn) / (1 - sn))
by A4, A5, A16, TOPRNS_1:25, XCMPLX_1:90
.=
|.p2.| * ((p2 `2 ) / |.p2.|)
by A12, XCMPLX_1:90
.=
p2 `2
by A4, TOPRNS_1:25, XCMPLX_1:88
;
then A19:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (1 - (((((|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]| `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)) ^2 ))) =
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (1 - (((p2 `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.|) ^2 )))
by A4, A16, TOPRNS_1:25, XCMPLX_1:90
.=
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (1 - (((p2 `2 ) ^2 ) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 ))))
by XCMPLX_1:77
.=
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (((|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 ) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 )) - (((p2 `2 ) ^2 ) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 ))))
by A10, A15, XCMPLX_1:60
.=
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (((|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 ) - ((p2 `2 ) ^2 )) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 )))
by XCMPLX_1:121
.=
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (((((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) - ((p2 `2 ) ^2 )) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 )))
by A15, JGRAPH_3:10
.=
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (((p2 `1 ) / |.p2.|) ^2 ))
by A16, XCMPLX_1:77
;
(p2 `1 ) / |.p2.| >= 0
by A4;
then A20:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (((p2 `1 ) / |.p2.|) ^2 )) =
|.p2.| * ((p2 `1 ) / |.p2.|)
by A16, SQUARE_1:89
.=
p2 `1
by A4, TOPRNS_1:25, XCMPLX_1:88
;
dom (sn -FanMorphE ) = the
carrier of
(TOP-REAL 2)
by FUNCT_2:def 1;
hence
ex
x being
set st
(
x in dom (sn -FanMorphE ) &
y = (sn -FanMorphE ) . x )
by A17, A18, A19, A20, EUCLID:57;
:: thesis: verum end; suppose A21:
(
(p2 `2 ) / |.p2.| < 0 &
p2 `1 >= 0 &
p2 <> 0. (TOP-REAL 2) )
;
:: thesis: ex x being set st
( x in dom (sn -FanMorphE ) & y = (sn -FanMorphE ) . x )set px =
|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|;
A22:
(
|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]| `1 = |.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 ))) &
|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]| `2 = |.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) )
by EUCLID:56;
then A23:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 =
((|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))) ^2 ) + ((|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn)) ^2 )
by JGRAPH_3:10
.=
((|.p2.| ^2 ) * ((sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 ))) ^2 )) + ((|.p2.| ^2 ) * (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 ))
;
A24:
|.p2.| <> 0
by A21, TOPRNS_1:25;
1
+ sn >= 0
by A1, XREAL_1:150;
then A25:
((p2 `2 ) / |.p2.|) * (1 + sn) <= 0
by A21;
then A26:
(((p2 `2 ) / |.p2.|) * (1 + sn)) + sn <= 0 + sn
by XREAL_1:9;
A27:
|.p2.| ^2 > 0
by A24, SQUARE_1:74;
1
- sn > 0
by A1, XREAL_1:151;
then A28:
(1 - sn) + sn >= (((p2 `2 ) / |.p2.|) * (1 + sn)) + sn
by A25, XREAL_1:9;
A29:
1
+ sn > 0
by A1, XREAL_1:150;
A30:
|.p2.| ^2 = ((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )
by JGRAPH_3:10;
0 <= (p2 `1 ) ^2
by XREAL_1:65;
then
0 + ((p2 `2 ) ^2 ) <= ((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )
by XREAL_1:9;
then
((p2 `2 ) ^2 ) / (|.p2.| ^2 ) <= (|.p2.| ^2 ) / (|.p2.| ^2 )
by A30, XREAL_1:74;
then
((p2 `2 ) ^2 ) / (|.p2.| ^2 ) <= 1
by A27, XCMPLX_1:60;
then
((p2 `2 ) / |.p2.|) ^2 <= 1
by XCMPLX_1:77;
then
(p2 `2 ) / |.p2.| >= - 1
by SQUARE_1:121;
then
((p2 `2 ) / |.p2.|) * (1 + sn) >= (- 1) * (1 + sn)
by A29, XREAL_1:66;
then
((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) - sn >= (- 1) - sn
;
then
(((p2 `2 ) / |.p2.|) * (1 + sn)) + sn >= - 1
by XREAL_1:11;
then
1
^2 >= ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2
by A28, SQUARE_1:119;
then A31:
1
- (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 ) >= 0
by XREAL_1:50;
then A32:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 =
((|.p2.| ^2 ) * (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 ))) + ((|.p2.| ^2 ) * (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 ))
by A23, SQUARE_1:def 4
.=
|.p2.| ^2
;
then A33:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| =
sqrt (|.p2.| ^2 )
by SQUARE_1:89
.=
|.p2.|
by SQUARE_1:89
;
sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )) >= 0
by A31, SQUARE_1:def 4;
then
(
(|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]| `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| <= sn &
|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]| `1 >= 0 &
|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]| <> 0. (TOP-REAL 2) )
by A22, A24, A26, A33, TOPRNS_1:24, XCMPLX_1:90;
then A34:
(sn -FanMorphE ) . |[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]| = |[(|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (1 - (((((|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]| `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)) ^2 )))),(|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| * ((((|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]| `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)))]|
by A1, Th91;
A35:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| * ((((|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]| `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)) =
|.p2.| * ((((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) - sn) / (1 + sn))
by A21, A22, A33, TOPRNS_1:25, XCMPLX_1:90
.=
|.p2.| * ((p2 `2 ) / |.p2.|)
by A29, XCMPLX_1:90
.=
p2 `2
by A21, TOPRNS_1:25, XCMPLX_1:88
;
then A36:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (1 - (((((|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]| `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)) ^2 ))) =
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (1 - (((p2 `2 ) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.|) ^2 )))
by A21, A33, TOPRNS_1:25, XCMPLX_1:90
.=
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (1 - (((p2 `2 ) ^2 ) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 ))))
by XCMPLX_1:77
.=
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (((|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 ) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 )) - (((p2 `2 ) ^2 ) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 ))))
by A27, A32, XCMPLX_1:60
.=
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (((|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 ) - ((p2 `2 ) ^2 )) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 )))
by XCMPLX_1:121
.=
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (((((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 )) - ((p2 `2 ) ^2 )) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 )))
by A32, JGRAPH_3:10
.=
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (((p2 `1 ) / |.p2.|) ^2 ))
by A33, XCMPLX_1:77
;
(p2 `1 ) / |.p2.| >= 0
by A21;
then A37:
|.|[(|.p2.| * (sqrt (1 - (((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn) ^2 )))),(|.p2.| * ((((p2 `2 ) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (((p2 `1 ) / |.p2.|) ^2 )) =
|.p2.| * ((p2 `1 ) / |.p2.|)
by A33, SQUARE_1:89
.=
p2 `1
by A21, TOPRNS_1:25, XCMPLX_1:88
;
dom (sn -FanMorphE ) = the
carrier of
(TOP-REAL 2)
by FUNCT_2:def 1;
hence
ex
x being
set st
(
x in dom (sn -FanMorphE ) &
y = (sn -FanMorphE ) . x )
by A34, A35, A36, A37, EUCLID:57;
:: thesis: verum end;
hence
y in rng f
by A2, FUNCT_1:def 5;
:: thesis: verum
end;
hence
rng (sn -FanMorphE ) = the
carrier of
(TOP-REAL 2)
by A2, XBOOLE_0:def 10;
:: thesis: verum
end;
hence
rng (sn -FanMorphE ) = the carrier of (TOP-REAL 2)
; :: thesis: verum