let sn be Real; :: thesis: for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
(sn -FanMorphE) . x in K0
let x, K0 be set ; :: thesis: ( - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } implies (sn -FanMorphE) . x in K0 )
assume A1: ( - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; :: thesis: (sn -FanMorphE) . x in K0
then consider p being Point of (TOP-REAL 2) such that
A2: p = x and
A3: p `1 >= 0 and
A4: p <> 0. (TOP-REAL 2) ;
A5: now :: thesis: not |.p.| <= 0
assume |.p.| <= 0 ; :: thesis: contradiction
then |.p.| = 0 ;
hence contradiction by A4, TOPRNS_1:24; :: thesis: verum
end;
then A6: |.p.| ^2 > 0 by SQUARE_1:12;
per cases ( (p `2) / |.p.| <= sn or (p `2) / |.p.| > sn ) ;
suppose A7: (p `2) / |.p.| <= sn ; :: thesis: (sn -FanMorphE) . x in K0
reconsider p9 = (sn -FanMorphE) . p as Point of (TOP-REAL 2) ;
(sn -FanMorphE) . p = |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| by A1, A3, A4, A7, Th84;
then A8: p9 `1 = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) by EUCLID:52;
A9: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1;
A10: 1 + sn > 0 by A1, XREAL_1:148;
per cases ( p `1 = 0 or p `1 <> 0 ) ;
suppose p `1 <> 0 ; :: thesis: (sn -FanMorphE) . x in K0
then 0 + ((p `2) ^2) < ((p `1) ^2) + ((p `2) ^2) by SQUARE_1:12, XREAL_1:8;
then ((p `2) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A6, A9, XREAL_1:74;
then ((p `2) ^2) / (|.p.| ^2) < 1 by A6, XCMPLX_1:60;
then ((p `2) / |.p.|) ^2 < 1 by XCMPLX_1:76;
then - 1 < (p `2) / |.p.| by SQUARE_1:52;
then (- 1) - sn < ((p `2) / |.p.|) - sn by XREAL_1:9;
then ((- 1) * (1 + sn)) / (1 + sn) < (((p `2) / |.p.|) - sn) / (1 + sn) by A10, XREAL_1:74;
then A11: - 1 < (((p `2) / |.p.|) - sn) / (1 + sn) by A10, XCMPLX_1:89;
((p `2) / |.p.|) - sn <= 0 by A7, XREAL_1:47;
then 1 ^2 > ((((p `2) / |.p.|) - sn) / (1 + sn)) ^2 by A10, A11, SQUARE_1:50;
then 1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2) > 0 by XREAL_1:50;
then sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)) > 0 by SQUARE_1:25;
then |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) > 0 by A5, XREAL_1:129;
hence (sn -FanMorphE) . x in K0 by A1, A2, A8, JGRAPH_2:3; :: thesis: verum
end;
end;
end;
suppose A12: (p `2) / |.p.| > sn ; :: thesis: (sn -FanMorphE) . x in K0
reconsider p9 = (sn -FanMorphE) . p as Point of (TOP-REAL 2) ;
(sn -FanMorphE) . p = |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| by A1, A3, A4, A12, Th84;
then A13: p9 `1 = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) by EUCLID:52;
A14: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1;
A15: 1 - sn > 0 by A1, XREAL_1:149;
per cases ( p `1 = 0 or p `1 <> 0 ) ;
suppose p `1 <> 0 ; :: thesis: (sn -FanMorphE) . x in K0
then 0 + ((p `2) ^2) < ((p `1) ^2) + ((p `2) ^2) by SQUARE_1:12, XREAL_1:8;
then ((p `2) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A6, A14, XREAL_1:74;
then ((p `2) ^2) / (|.p.| ^2) < 1 by A6, XCMPLX_1:60;
then ((p `2) / |.p.|) ^2 < 1 by XCMPLX_1:76;
then (p `2) / |.p.| < 1 by SQUARE_1:52;
then ((p `2) / |.p.|) - sn < 1 - sn by XREAL_1:9;
then (((p `2) / |.p.|) - sn) / (1 - sn) < (1 - sn) / (1 - sn) by A15, XREAL_1:74;
then A16: (((p `2) / |.p.|) - sn) / (1 - sn) < 1 by A15, XCMPLX_1:60;
( - (1 - sn) < - 0 & ((p `2) / |.p.|) - sn >= sn - sn ) by A12, A15, XREAL_1:9, XREAL_1:24;
then ((- 1) * (1 - sn)) / (1 - sn) < (((p `2) / |.p.|) - sn) / (1 - sn) by A15, XREAL_1:74;
then - 1 < (((p `2) / |.p.|) - sn) / (1 - sn) by A15, XCMPLX_1:89;
then 1 ^2 > ((((p `2) / |.p.|) - sn) / (1 - sn)) ^2 by A16, SQUARE_1:50;
then 1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2) > 0 by XREAL_1:50;
then sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)) > 0 by SQUARE_1:25;
then p9 `1 > 0 by A5, A13, XREAL_1:129;
hence (sn -FanMorphE) . x in K0 by A1, A2, JGRAPH_2:3; :: thesis: verum
end;
end;
end;
end;