let cn be Real; :: thesis: 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 ; :: thesis: ( - 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) ) } ) ; :: thesis: (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) ;
A5: now
assume |.p.| <= 0 ; :: thesis: contradiction
then |.p.| = 0 ;
hence contradiction by A4, TOPRNS_1:25; :: thesis: verum
end;
then A6: |.p.| ^2 > 0 by SQUARE_1:74;
per cases ( (p `1 ) / |.p.| <= cn or (p `1 ) / |.p.| > cn ) ;
suppose A7: (p `1 ) / |.p.| <= cn ; :: thesis: (cn -FanMorphN ) . x in K0
reconsider 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, Th58;
then A8: p9 `2 = |.p.| * (sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))) by EUCLID:56;
A9: |.p.| ^2 = ((p `1 ) ^2 ) + ((p `2 ) ^2 ) by JGRAPH_3:10;
A10: 1 + cn > 0 by A1, XREAL_1:150;
per cases ( p `2 = 0 or p `2 <> 0 ) ;
suppose p `2 <> 0 ; :: thesis: (cn -FanMorphN ) . x in K0
then 0 + ((p `1 ) ^2 ) < ((p `1 ) ^2 ) + ((p `2 ) ^2 ) by SQUARE_1:74, XREAL_1:10;
then ((p `1 ) ^2 ) / (|.p.| ^2 ) < (|.p.| ^2 ) / (|.p.| ^2 ) by A6, A9, XREAL_1:76;
then ((p `1 ) ^2 ) / (|.p.| ^2 ) < 1 by A6, XCMPLX_1:60;
then ((p `1 ) / |.p.|) ^2 < 1 by XCMPLX_1:77;
then - 1 < (p `1 ) / |.p.| by SQUARE_1:122;
then (- 1) - cn < ((p `1 ) / |.p.|) - cn by XREAL_1:11;
then ((- 1) * (1 + cn)) / (1 + cn) < (((p `1 ) / |.p.|) - cn) / (1 + cn) by A10, XREAL_1:76;
then A11: - 1 < (((p `1 ) / |.p.|) - cn) / (1 + cn) by A10, XCMPLX_1:90;
((p `1 ) / |.p.|) - cn <= 0 by A7, XREAL_1:49;
then 1 ^2 > ((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 by A10, A11, SQUARE_1:120;
then 1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ) > 0 by XREAL_1:52;
then sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 )) > 0 by SQUARE_1:93;
then |.p.| * (sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 + cn)) ^2 ))) > 0 by A5, XREAL_1:131;
hence (cn -FanMorphN ) . x in K0 by A1, A2, A8, JGRAPH_2:11; :: thesis: verum
end;
end;
end;
suppose A12: (p `1 ) / |.p.| > cn ; :: thesis: (cn -FanMorphN ) . x in K0
reconsider 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, Th58;
then A13: p9 `2 = |.p.| * (sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ))) by EUCLID:56;
A14: |.p.| ^2 = ((p `1 ) ^2 ) + ((p `2 ) ^2 ) by JGRAPH_3:10;
A15: 1 - cn > 0 by A1, XREAL_1:151;
per cases ( p `2 = 0 or p `2 <> 0 ) ;
suppose p `2 <> 0 ; :: thesis: (cn -FanMorphN ) . x in K0
then 0 + ((p `1 ) ^2 ) < ((p `1 ) ^2 ) + ((p `2 ) ^2 ) by SQUARE_1:74, XREAL_1:10;
then ((p `1 ) ^2 ) / (|.p.| ^2 ) < (|.p.| ^2 ) / (|.p.| ^2 ) by A6, A14, XREAL_1:76;
then ((p `1 ) ^2 ) / (|.p.| ^2 ) < 1 by A6, XCMPLX_1:60;
then ((p `1 ) / |.p.|) ^2 < 1 by XCMPLX_1:77;
then (p `1 ) / |.p.| < 1 by SQUARE_1:122;
then ((p `1 ) / |.p.|) - cn < 1 - cn by XREAL_1:11;
then (((p `1 ) / |.p.|) - cn) / (1 - cn) < (1 - cn) / (1 - cn) by A15, XREAL_1:76;
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_1:11, XREAL_1:26;
then ((- 1) * (1 - cn)) / (1 - cn) < (((p `1 ) / |.p.|) - cn) / (1 - cn) by A15, XREAL_1:76;
then - 1 < (((p `1 ) / |.p.|) - cn) / (1 - cn) by A15, XCMPLX_1:90;
then 1 ^2 > ((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 by A16, SQUARE_1:120;
then 1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 ) > 0 by XREAL_1:52;
then sqrt (1 - (((((p `1 ) / |.p.|) - cn) / (1 - cn)) ^2 )) > 0 by SQUARE_1:93;
then p9 `2 > 0 by A5, A13, XREAL_1:131;
hence (cn -FanMorphN ) . x in K0 by A1, A2, JGRAPH_2:11; :: thesis: verum
end;
end;
end;
end;