let cn be Real; for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous
let K1 be non empty Subset of (TOP-REAL 2); for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous
let f be Function of ((TOP-REAL 2) | K1),R^1; ( - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) implies f is continuous )
reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5;
set a = cn;
set b = 1 + cn;
reconsider g2 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm2;
assume that
A1:
- 1 < cn
and
A2:
for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))
and
A3:
for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) )
; f is continuous
A4:
1 + cn > 0
by A1, XREAL_1:148;
for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q <> 0. (TOP-REAL 2)
by A3;
then
for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0
by Lm6;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A5:
for q being Point of ((TOP-REAL 2) | K1)
for r1, r2 being Real st g2 . q = r1 & g1 . q = r2 holds
g3 . q = r2 * (sqrt |.(1 - ((((r1 / r2) - cn) / (1 + cn)) ^2)).|)
and
A6:
g3 is continuous
by A4, Th10;
A7:
dom g3 = the carrier of ((TOP-REAL 2) | K1)
by FUNCT_2:def 1;
then A8:
dom f = dom g3
by FUNCT_2:def 1;
for x being object st x in dom f holds
f . x = g3 . x
proof
let x be
object ;
( x in dom f implies f . x = g3 . x )
assume A9:
x in dom f
;
f . x = g3 . x
then
x in K1
by A7, A8, PRE_TOPC:8;
then reconsider r =
x as
Point of
(TOP-REAL 2) ;
reconsider s =
x as
Point of
((TOP-REAL 2) | K1) by A9;
A10:
(1 + cn) ^2 > 0
by A4, SQUARE_1:12;
A11:
|.r.| <> 0
by A3, A9, TOPRNS_1:24;
|.r.| ^2 = ((r `1) ^2) + ((r `2) ^2)
by JGRAPH_3:1;
then A12:
((r `1) - |.r.|) * ((r `1) + |.r.|) = - ((r `2) ^2)
;
(r `2) ^2 >= 0
by XREAL_1:63;
then
- |.r.| <= r `1
by A12, XREAL_1:93;
then
(r `1) / |.r.| >= (- |.r.|) / |.r.|
by XREAL_1:72;
then
(r `1) / |.r.| >= - 1
by A11, XCMPLX_1:197;
then
((r `1) / |.r.|) - cn >= (- 1) - cn
by XREAL_1:9;
then A13:
((r `1) / |.r.|) - cn >= - (1 + cn)
;
cn - ((r `1) / |.r.|) >= 0
by A3, A9, XREAL_1:48;
then
- (cn - ((r `1) / |.r.|)) <= - 0
;
then
(((r `1) / |.r.|) - cn) ^2 <= (1 + cn) ^2
by A4, A13, SQUARE_1:49;
then
((((r `1) / |.r.|) - cn) ^2) / ((1 + cn) ^2) <= ((1 + cn) ^2) / ((1 + cn) ^2)
by A4, XREAL_1:72;
then
((((r `1) / |.r.|) - cn) ^2) / ((1 + cn) ^2) <= 1
by A10, XCMPLX_1:60;
then
((((r `1) / |.r.|) - cn) / (1 + cn)) ^2 <= 1
by XCMPLX_1:76;
then
1
- (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2) >= 0
by XREAL_1:48;
then
|.(1 - (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2)).| = 1
- (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2)
by ABSVALUE:def 1;
then A14:
f . r = |.r.| * (sqrt |.(1 - (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2)).|)
by A2, A9;
A15:
(
proj1 . r = r `1 &
(2 NormF) . r = |.r.| )
by Def1, PSCOMP_1:def 5;
(
g2 . s = proj1 . s &
g1 . s = (2 NormF) . s )
by Lm2, Lm5;
hence
f . x = g3 . x
by A5, A14, A15;
verum
end;
hence
f is continuous
by A6, A8, FUNCT_1:2; verum