let cn 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 < cn & cn < 1 & f = (cn -FanMorphS ) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 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 < cn & cn < 1 & f = (cn -FanMorphS ) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 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 < cn & cn < 1 & f = (cn -FanMorphS ) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous )
assume A1: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS ) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; :: thesis: f is continuous
set sn = sqrt (1 - (cn ^2 ));
set p0 = |[cn,(sqrt (1 - (cn ^2 )))]|;
A2: |[cn,(sqrt (1 - (cn ^2 )))]| `2 = sqrt (1 - (cn ^2 )) by EUCLID:56;
cn ^2 < 1 ^2 by A1, SQUARE_1:120;
then A3: 1 - (cn ^2 ) > 0 by XREAL_1:52;
then A4: sqrt (1 - (cn ^2 )) > 0 by SQUARE_1:93;
A5: |[cn,(sqrt (1 - (cn ^2 )))]| `2 > 0 by A2, A3, SQUARE_1:93;
|[cn,(sqrt (1 - (cn ^2 )))]| in K0 by A1, A2, A4, JGRAPH_2:11;
then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ;
( |[cn,(sqrt (1 - (cn ^2 )))]| in the carrier of (TOP-REAL 2) & not |[cn,(sqrt (1 - (cn ^2 )))]| in {(0. (TOP-REAL 2))} ) by A5, JGRAPH_2:11, TARSKI:def 1;
then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A1, XBOOLE_0:def 5;
A6: 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 p2 being Point of (TOP-REAL 2) such that
A7: ( p2 = x & p2 `2 >= 0 & p2 <> 0. (TOP-REAL 2) ) by A1;
not p2 in {(0. (TOP-REAL 2))} by A7, TARSKI:def 1;
hence x in D by A1, A7, XBOOLE_0:def 5; :: thesis: verum

end;

for p being Point of ((TOP-REAL 2) | K1)
for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds
ex W being Subset of ((TOP-REAL 2) | K1) st
( p in W & W is open & f .: W c= V )

proof

let p be Point of ((TOP-REAL 2) | K1); :: thesis: for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds
ex W being Subset of ((TOP-REAL 2) | K1) st
( p in W & W is open & f .: W c= V )
let V be Subset of ((TOP-REAL 2) | D); :: thesis: ( f . p in V & V is open implies ex W being Subset of ((TOP-REAL 2) | K1) st
( p in W & W is open & f .: W c= V ) )
assume A8: ( f . p in V & V is open ) ; :: thesis: ex W being Subset of ((TOP-REAL 2) | K1) st
( p in W & W is open & f .: W c= V )
then consider V2 being Subset of (TOP-REAL 2) such that
A9: ( V2 is open & V2 /\ ([#] ((TOP-REAL 2) | D)) = V ) by TOPS_2:32;
A10: p in the carrier of ((TOP-REAL 2) | K1) ;
A11: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def 10;
reconsider W2 = V2 /\ ([#] ((TOP-REAL 2) | K1)) as Subset of ((TOP-REAL 2) | K1) ;
A12: W2 is open by A9, TOPS_2:32;
A13: f . p = (cn -FanMorphS ) . p by A1, A11, FUNCT_1:72;
consider q being Point of (TOP-REAL 2) such that
A14: ( q = p & q `2 >= 0 & q <> 0. (TOP-REAL 2) ) by A1, A10, A11;
(cn -FanMorphS ) . q = q by A14, Th120;
then ( p in V2 & p in [#] ((TOP-REAL 2) | D) ) by A8, A9, A13, A14, XBOOLE_0:def 4;
then A15: p in W2 by XBOOLE_0:def 4;
f .: W2 c= V

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in f .: W2 or y in V )
assume y in f .: W2 ; :: thesis: y in V
then consider x being set such that
A16: ( x in dom f & x in W2 & y = f . x ) by FUNCT_1:def 12;
f is Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) ;
then dom f = K1 by A11, FUNCT_2:def 1;
then consider p4 being Point of (TOP-REAL 2) such that
A17: ( x = p4 & p4 `2 >= 0 & p4 <> 0. (TOP-REAL 2) ) by A1, A16;
A18: f . p4 = (cn -FanMorphS ) . p4 by A1, A11, A16, A17, FUNCT_1:72
.= p4 by A17, Th120 ;
A19: ( p4 in V2 & p4 in [#] ((TOP-REAL 2) | K1) ) by A16, A17, XBOOLE_0:def 4;
then p4 in D by A6, A11;
then p4 in [#] ((TOP-REAL 2) | D) by PRE_TOPC:def 10;
hence y in V by A9, A16, A17, A18, A19, XBOOLE_0:def 4; :: thesis: verum

end;

hence ex W being Subset of ((TOP-REAL 2) | K1) st
( p in W & W is open & f .: W c= V ) by A12, A15; :: thesis: verum

end;

hence f is continuous by JGRAPH_2:20; :: thesis: verum