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 -FanMorphN ) | 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 -FanMorphN ) | 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 -FanMorphN ) | 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 )
set sn = sqrt (1 - (cn ^2 ));
set p0 = |[cn,(sqrt (1 - (cn ^2 )))]|;
A1: |[cn,(sqrt (1 - (cn ^2 )))]| `2 = sqrt (1 - (cn ^2 )) by EUCLID:56;
assume A2: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN ) | 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
then cn ^2 < 1 ^2 by SQUARE_1:120;
then A3: 1 - (cn ^2 ) > 0 by XREAL_1:52;
then A4: |[cn,(sqrt (1 - (cn ^2 )))]| `2 > 0 by A1, SQUARE_1:93;
then |[cn,(sqrt (1 - (cn ^2 )))]| in K0 by A2, JGRAPH_2:11;
then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ;
|[cn,(sqrt (1 - (cn ^2 )))]| <> 0. (TOP-REAL 2) by A1, A3, JGRAPH_2:11, SQUARE_1:93;
then not |[cn,(sqrt (1 - (cn ^2 )))]| in {(0. (TOP-REAL 2))} by TARSKI:def 1;
then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A2, XBOOLE_0:def 5;
A5: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
|[cn,(sqrt (1 - (cn ^2 )))]| `1 = cn by EUCLID:56;
then A6: |.|[cn,(sqrt (1 - (cn ^2 )))]|.| = sqrt (((sqrt (1 - (cn ^2 ))) ^2 ) + (cn ^2 )) by A1, JGRAPH_3:10;
A7: D <> {} ;
(sqrt (1 - (cn ^2 ))) ^2 = 1 - (cn ^2 ) by A3, SQUARE_1:def 4;
then A8: (|[cn,(sqrt (1 - (cn ^2 )))]| `1 ) / |.|[cn,(sqrt (1 - (cn ^2 )))]|.| = cn by A6, EUCLID:56, SQUARE_1:83;
then A9: |[cn,(sqrt (1 - (cn ^2 )))]| in { p where p is Point of (TOP-REAL 2) : ( (p `1 ) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } by A4, JGRAPH_2:11;
A10: { p where p is Point of (TOP-REAL 2) : ( (p `1 ) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-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 `1 ) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 )
assume x in { p where p is Point of (TOP-REAL 2) : ( (p `1 ) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ; :: thesis: x in K1
then ex p being Point of (TOP-REAL 2) st
( p = x & (p `1 ) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) ;
hence x in K1 by A2; :: thesis: verum

end;

A11: { p where p is Point of (TOP-REAL 2) : ( (p `1 ) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-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 `1 ) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 )
assume x in { p where p is Point of (TOP-REAL 2) : ( (p `1 ) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ; :: thesis: x in K1
then ex p being Point of (TOP-REAL 2) st
( p = x & (p `1 ) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) ;
hence x in K1 by A2; :: thesis: verum

end;

then reconsider K00 = { p where p is Point of (TOP-REAL 2) : ( (p `1 ) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A9, PRE_TOPC:29;
the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:29;
then A12: rng (f | K00) c= D ;
|[cn,(sqrt (1 - (cn ^2 )))]| in { p where p is Point of (TOP-REAL 2) : ( (p `1 ) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } by A4, A8, JGRAPH_2:11;
then reconsider K11 = { p where p is Point of (TOP-REAL 2) : ( (p `1 ) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A10, PRE_TOPC:29;
the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:29;
then A13: rng (f | K11) c= D ;
the carrier of ((TOP-REAL 2) | B0) = the carrier of ((TOP-REAL 2) | D) ;
then A14: dom f = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def 1
.= K1 by PRE_TOPC:29 ;
then dom (f | K00) = K00 by A11, RELAT_1:91
.= the carrier of (((TOP-REAL 2) | K1) | K00) by PRE_TOPC:29 ;
then reconsider f1 = f | K00 as Function of (((TOP-REAL 2) | K1) | K00),((TOP-REAL 2) | D) by A12, FUNCT_2:4;
dom (f | K11) = K11 by A10, A14, RELAT_1:91
.= the carrier of (((TOP-REAL 2) | K1) | K11) by PRE_TOPC:29 ;
then reconsider f2 = f | K11 as Function of (((TOP-REAL 2) | K1) | K11),((TOP-REAL 2) | D) by A13, FUNCT_2:4;
defpred S₁[ Point of (TOP-REAL 2)] means ( ($1 `1 ) / |.$1.| >= cn & $1 `2 >= 0 & $1 <> 0. (TOP-REAL 2) );
A15: dom f2 = the carrier of (((TOP-REAL 2) | K1) | K11) by FUNCT_2:def 1
.= K11 by PRE_TOPC:29 ;
{ p where p is Point of (TOP-REAL 2) : S₁[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch 7();
then reconsider K001 = { p where p is Point of (TOP-REAL 2) : ( (p `1 ) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A9;
A16: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
defpred S₂[ Point of (TOP-REAL 2)] means ( $1 `1 >= cn * |.$1.| & $1 `2 >= 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 `1 >= cn * |.p.| & p `2 >= 0 ) } as Subset of (TOP-REAL 2) ;
defpred S₃[ Point of (TOP-REAL 2)] means ( ($1 `1 ) / |.$1.| <= cn & $1 `2 >= 0 & $1 <> 0. (TOP-REAL 2) );
A17: { p where p is Point of (TOP-REAL 2) : S₃[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch 7();
A18: rng ((cn -FanMorphN ) | K001) c= K1

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ((cn -FanMorphN ) | K001) or y in K1 )
assume y in rng ((cn -FanMorphN ) | K001) ; :: thesis: y in K1
then consider x being set such that
A19: x in dom ((cn -FanMorphN ) | K001) and
A20: y = ((cn -FanMorphN ) | K001) . x by FUNCT_1:def 5;
reconsider q = x as Point of (TOP-REAL 2) by A19;
A21: y = (cn -FanMorphN ) . q by A19, A20, FUNCT_1:70;
dom ((cn -FanMorphN ) | K001) = (dom (cn -FanMorphN )) /\ K001 by RELAT_1:90
.= the carrier of (TOP-REAL 2) /\ K001 by FUNCT_2:def 1
.= K001 by XBOOLE_1:28 ;
then A22: ex p2 being Point of (TOP-REAL 2) st
( p2 = q & (p2 `1 ) / |.p2.| >= cn & p2 `2 >= 0 & p2 <> 0. (TOP-REAL 2) ) by A19;
then A23: ((q `1 ) / |.q.|) - cn >= 0 by XREAL_1:50;
|.q.| <> 0 by A22, TOPRNS_1:25;
then A24: |.q.| ^2 > 0 ^2 by SQUARE_1:74;
set q4 = |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]|;
A25: |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| `1 = |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn)) by EUCLID:56;
A26: 1 - cn > 0 by A2, XREAL_1:151;
0 <= (q `2 ) ^2 by XREAL_1:65;
then 0 + ((q `1 ) ^2 ) <= ((q `1 ) ^2 ) + ((q `2 ) ^2 ) by XREAL_1:9;
then (q `1 ) ^2 <= |.q.| ^2 by JGRAPH_3:10;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) <= (|.q.| ^2 ) / (|.q.| ^2 ) by XREAL_1:74;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) <= 1 by A24, XCMPLX_1:60;
then ((q `1 ) / |.q.|) ^2 <= 1 by XCMPLX_1:77;
then 1 >= (q `1 ) / |.q.| by SQUARE_1:121;
then 1 - cn >= ((q `1 ) / |.q.|) - cn by XREAL_1:11;
then - (1 - cn) <= - (((q `1 ) / |.q.|) - cn) by XREAL_1:26;
then (- (1 - cn)) / (1 - cn) <= (- (((q `1 ) / |.q.|) - cn)) / (1 - cn) by A26, XREAL_1:74;
then - 1 <= (- (((q `1 ) / |.q.|) - cn)) / (1 - cn) by A26, XCMPLX_1:198;
then ((- (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A26, A23, SQUARE_1:119;
then A27: 1 - (((- (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) >= 0 by XREAL_1:50;
then A28: 1 - ((- ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 ) >= 0 by XCMPLX_1:188;
sqrt (1 - (((- (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 )) >= 0 by A27, SQUARE_1:def 4;
then sqrt (1 - (((- (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 - cn) ^2 ))) >= 0 by XCMPLX_1:77;
then sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 - cn) ^2 ))) >= 0 ;
then A29: sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )) >= 0 by XCMPLX_1:77;
A30: |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| `2 = |.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) by EUCLID:56;
then A31: (|[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| `2 ) ^2 = (|.q.| ^2 ) * ((sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) ^2 )
.= (|.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )) by A28, SQUARE_1:def 4 ;
|.|[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]|.| ^2 = ((|[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| `1 ) ^2 ) + ((|[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| `2 ) ^2 ) by JGRAPH_3:10
.= |.q.| ^2 by A25, A31 ;
then A32: |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| <> 0. (TOP-REAL 2) by A24, TOPRNS_1:24;
(cn -FanMorphN ) . q = |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| by A2, A22, Th58;
hence y in K1 by A2, A21, A30, A29, A32; :: thesis: verum

end;

A33: dom (cn -FanMorphN ) = the carrier of (TOP-REAL 2) by FUNCT_2:def 1;
then dom ((cn -FanMorphN ) | K001) = K001 by RELAT_1:91
.= the carrier of ((TOP-REAL 2) | K001) by PRE_TOPC:29 ;
then reconsider f3 = (cn -FanMorphN ) | K001 as Function of ((TOP-REAL 2) | K001),((TOP-REAL 2) | K1) by A5, A18, FUNCT_2:4;
A34: K003 is closed by Th65;
K1 c= D

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in K1 or x in D )
assume A35: x in K1 ; :: thesis: x in D
then ex p6 being Point of (TOP-REAL 2) st
( p6 = x & p6 `2 >= 0 & p6 <> 0. (TOP-REAL 2) ) by A2;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def 1;
hence x in D by A2, A35, XBOOLE_0:def 5; :: thesis: verum

end;

then D = K1 \/ D by XBOOLE_1:12;
then A36: (TOP-REAL 2) | K1 is SubSpace of (TOP-REAL 2) | D by TOPMETR:5;
|[cn,(sqrt (1 - (cn ^2 )))]| in { p where p is Point of (TOP-REAL 2) : ( (p `1 ) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } by A4, A8, JGRAPH_2:11;
then reconsider K111 = { p where p is Point of (TOP-REAL 2) : ( (p `1 ) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A17;
A37: rng ((cn -FanMorphN ) | K111) c= K1

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ((cn -FanMorphN ) | K111) or y in K1 )
assume y in rng ((cn -FanMorphN ) | K111) ; :: thesis: y in K1
then consider x being set such that
A38: x in dom ((cn -FanMorphN ) | K111) and
A39: y = ((cn -FanMorphN ) | K111) . x by FUNCT_1:def 5;
reconsider q = x as Point of (TOP-REAL 2) by A38;
A40: y = (cn -FanMorphN ) . q by A38, A39, FUNCT_1:70;
dom ((cn -FanMorphN ) | K111) = (dom (cn -FanMorphN )) /\ K111 by RELAT_1:90
.= the carrier of (TOP-REAL 2) /\ K111 by FUNCT_2:def 1
.= K111 by XBOOLE_1:28 ;
then A41: ex p2 being Point of (TOP-REAL 2) st
( p2 = q & (p2 `1 ) / |.p2.| <= cn & p2 `2 >= 0 & p2 <> 0. (TOP-REAL 2) ) by A38;
then A42: ((q `1 ) / |.q.|) - cn <= 0 by XREAL_1:49;
|.q.| <> 0 by A41, TOPRNS_1:25;
then A43: |.q.| ^2 > 0 ^2 by SQUARE_1:74;
set q4 = |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]|;
A44: |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| `1 = |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by EUCLID:56;
A45: 1 + cn > 0 by A2, XREAL_1:150;
0 <= (q `2 ) ^2 by XREAL_1:65;
then ( |.q.| ^2 = ((q `1 ) ^2 ) + ((q `2 ) ^2 ) & 0 + ((q `1 ) ^2 ) <= ((q `1 ) ^2 ) + ((q `2 ) ^2 ) ) by JGRAPH_3:10, XREAL_1:9;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) <= (|.q.| ^2 ) / (|.q.| ^2 ) by XREAL_1:74;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) <= 1 by A43, XCMPLX_1:60;
then ((q `1 ) / |.q.|) ^2 <= 1 by XCMPLX_1:77;
then - 1 <= (q `1 ) / |.q.| by SQUARE_1:121;
then (- 1) - cn <= ((q `1 ) / |.q.|) - cn by XREAL_1:11;
then (- (1 + cn)) / (1 + cn) <= (((q `1 ) / |.q.|) - cn) / (1 + cn) by A45, XREAL_1:74;
then - 1 <= (((q `1 ) / |.q.|) - cn) / (1 + cn) by A45, XCMPLX_1:198;
then A46: ((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 <= 1 ^2 by A45, A42, SQUARE_1:119;
then A47: 1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ) >= 0 by XREAL_1:50;
1 - ((- ((((q `1 ) / |.q.|) - cn) / (1 + cn))) ^2 ) >= 0 by A46, XREAL_1:50;
then 1 - (((- (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 ) >= 0 by XCMPLX_1:188;
then sqrt (1 - (((- (((q `1 ) / |.q.|) - cn)) / (1 + cn)) ^2 )) >= 0 by SQUARE_1:def 4;
then sqrt (1 - (((- (((q `1 ) / |.q.|) - cn)) ^2 ) / ((1 + cn) ^2 ))) >= 0 by XCMPLX_1:77;
then sqrt (1 - (((((q `1 ) / |.q.|) - cn) ^2 ) / ((1 + cn) ^2 ))) >= 0 ;
then A48: sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )) >= 0 by XCMPLX_1:77;
A49: |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| `2 = |.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) by EUCLID:56;
then A50: (|[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| `2 ) ^2 = (|.q.| ^2 ) * ((sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) ^2 )
.= (|.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )) by A47, SQUARE_1:def 4 ;
|.|[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]|.| ^2 = ((|[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| `1 ) ^2 ) + ((|[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| `2 ) ^2 ) by JGRAPH_3:10
.= |.q.| ^2 by A44, A50 ;
then A51: |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| <> 0. (TOP-REAL 2) by A43, TOPRNS_1:24;
(cn -FanMorphN ) . q = |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| by A2, A41, Th58;
hence y in K1 by A2, A40, A49, A48, A51; :: thesis: verum

end;

dom ((cn -FanMorphN ) | K111) = K111 by A33, RELAT_1:91
.= the carrier of ((TOP-REAL 2) | K111) by PRE_TOPC:29 ;
then reconsider f4 = (cn -FanMorphN ) | K111 as Function of ((TOP-REAL 2) | K111),((TOP-REAL 2) | K1) by A16, A37, FUNCT_2:4;
the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
then ( ((TOP-REAL 2) | K1) | K11 = (TOP-REAL 2) | K111 & f2 = f4 ) by A2, FUNCT_1:82, GOBOARD9:4;
then A52: f2 is continuous by A2, A36, Th64, PRE_TOPC:56;
A53: the carrier of ((TOP-REAL 2) | K1) = K0 by PRE_TOPC:29;
set T1 = ((TOP-REAL 2) | K1) | K00;
set T2 = ((TOP-REAL 2) | K1) | K11;
A54: [#] (((TOP-REAL 2) | K1) | K11) = K11 by PRE_TOPC:def 10;
defpred S₄[ Point of (TOP-REAL 2)] means ( $1 `1 <= cn * |.$1.| & $1 `2 >= 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 `1 <= cn * |.p.| & p `2 >= 0 ) } as Subset of (TOP-REAL 2) ;
A55: 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 A56: x in K004 /\ K1 ; :: thesis: x in K11
then x in K004 by XBOOLE_0:def 4;
then consider q1 being Point of (TOP-REAL 2) such that
A57: q1 = x and
A58: q1 `1 <= cn * |.q1.| and
q1 `2 >= 0 ;
x in K1 by A56, XBOOLE_0:def 4;
then A59: ex q2 being Point of (TOP-REAL 2) st
( q2 = x & q2 `2 >= 0 & q2 <> 0. (TOP-REAL 2) ) by A2;
(q1 `1 ) / |.q1.| <= (cn * |.q1.|) / |.q1.| by A58, XREAL_1:74;
then (q1 `1 ) / |.q1.| <= cn by A57, A59, TOPRNS_1:25, XCMPLX_1:90;
hence x in K11 by A57, A59; :: thesis: verum

end;

A60: K004 is closed by Th66;
the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:29;
then ( ((TOP-REAL 2) | K1) | K00 = (TOP-REAL 2) | K001 & f1 = f3 ) by A2, FUNCT_1:82, GOBOARD9:4;
then A61: f1 is continuous by A2, A36, Th63, PRE_TOPC:56;
A62: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def 10;
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
A63: p = x and
A64: (p `1 ) / |.p.| <= cn and
A65: p `2 >= 0 and
A66: p <> 0. (TOP-REAL 2) ;
((p `1 ) / |.p.|) * |.p.| <= cn * |.p.| by A64, XREAL_1:66;
then p `1 <= cn * |.p.| by A66, TOPRNS_1:25, XCMPLX_1:88;
then A67: x in K004 by A63, A65;
x in K1 by A2, A63, A65, A66;
hence x in K004 /\ K1 by A67, XBOOLE_0:def 4; :: thesis: verum

end;

then K11 = K004 /\ ([#] ((TOP-REAL 2) | K1)) by A62, A55, XBOOLE_0:def 10;
then A68: K11 is closed by A60, PRE_TOPC:43;
A69: 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 A70: x in K003 /\ K1 ; :: thesis: x in K00
then x in K003 by XBOOLE_0:def 4;
then consider q1 being Point of (TOP-REAL 2) such that
A71: q1 = x and
A72: q1 `1 >= cn * |.q1.| and
q1 `2 >= 0 ;
x in K1 by A70, XBOOLE_0:def 4;
then A73: ex q2 being Point of (TOP-REAL 2) st
( q2 = x & q2 `2 >= 0 & q2 <> 0. (TOP-REAL 2) ) by A2;
(q1 `1 ) / |.q1.| >= (cn * |.q1.|) / |.q1.| by A72, XREAL_1:74;
then (q1 `1 ) / |.q1.| >= cn by A71, A73, TOPRNS_1:25, XCMPLX_1:90;
hence x in K00 by A71, A73; :: 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
A74: p = x and
A75: (p `1 ) / |.p.| >= cn and
A76: p `2 >= 0 and
A77: p <> 0. (TOP-REAL 2) ;
((p `1 ) / |.p.|) * |.p.| >= cn * |.p.| by A75, XREAL_1:66;
then p `1 >= cn * |.p.| by A77, TOPRNS_1:25, XCMPLX_1:88;
then A78: x in K003 by A74, A76;
x in K1 by A2, A74, A76, A77;
hence x in K003 /\ K1 by A78, XBOOLE_0:def 4; :: thesis: verum

end;

then K00 = K003 /\ ([#] ((TOP-REAL 2) | K1)) by A62, A69, XBOOLE_0:def 10;
then A79: K00 is closed by A34, PRE_TOPC:43;
A80: [#] (((TOP-REAL 2) | K1) | K00) = K00 by PRE_TOPC:def 10;
A81: 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 A82: p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) ; :: thesis: f1 . p = f2 . p
then p in K00 by A80, XBOOLE_0:def 4;
hence f1 . p = f . p by FUNCT_1:72
.= f2 . p by A54, A82, FUNCT_1:72 ;
:: thesis: verum

end;

A83: 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
A84: ( p = x & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) by A2;

per cases ( (p `1 ) / |.p.| >= cn or (p `1 ) / |.p.| < cn ) ;

suppose (p `1 ) / |.p.| >= cn ; :: thesis: x in K00 \/ K11

then x in K00 by A84;
hence x in K00 \/ K11 by XBOOLE_0:def 3; :: thesis: verum

end;

suppose (p `1 ) / |.p.| < cn ; :: thesis: x in K00 \/ K11

then x in K11 by A84;
hence x in K00 \/ K11 by XBOOLE_0:def 3; :: thesis: verum

end;

end;

end;

then ([#] (((TOP-REAL 2) | K1) | K00)) \/ ([#] (((TOP-REAL 2) | K1) | K11)) = [#] ((TOP-REAL 2) | K1) by A80, A54, A62, XBOOLE_0:def 10;
then consider h being Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) such that
A85: h = f1 +* f2 and
A86: h is continuous by A80, A54, A79, A68, A61, A52, A81, JGRAPH_2:9;
A87: dom h = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def 1;
A88: dom f1 = the carrier of (((TOP-REAL 2) | K1) | K00) by FUNCT_2:def 1
.= K00 by PRE_TOPC:29 ;
A89: 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 A90: y in dom h ; :: thesis: h . y = f . y

per cases ( ( y in K00 & not y in K11 ) or y in K11 ) by A83, A87, A53, A90, XBOOLE_0:def 3;

suppose A91: ( y in K00 & not y in K11 ) ; :: thesis: h . y = f . y

then y in (dom f1) \/ (dom f2) by A88, XBOOLE_0:def 3;
hence h . y = f1 . y by A15, A85, A91, FUNCT_4:def 1
.= f . y by A91, FUNCT_1:72 ;
:: thesis: verum

end;

suppose A92: y in K11 ; :: thesis: h . y = f . y

then y in (dom f1) \/ (dom f2) by A15, XBOOLE_0:def 3;
hence h . y = f2 . y by A15, A85, A92, FUNCT_4:def 1
.= f . y by A92, FUNCT_1:72 ;
:: thesis: verum

end;

end;

end;

K0 = the carrier of ((TOP-REAL 2) | K0) by PRE_TOPC:29
.= dom f by A7, FUNCT_2:def 1 ;
hence f is continuous by A86, A87, A89, FUNCT_1:9, PRE_TOPC:29; :: thesis: verum