let q be Point of (TOP-REAL 2); :: thesis: for cn being Real st - 1 < cn & cn < 1 holds
( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) )
let cn be Real; :: thesis: ( - 1 < cn & cn < 1 implies ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) )
assume that
A1: - 1 < cn and
A2: cn < 1 ; :: thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) )

per cases ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) ) or ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) ) or q `2 > 0 or q = 0. (TOP-REAL 2) ) ;

suppose A3: ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ; :: thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) )

per cases ( q `2 < 0 or q `2 >= 0 ) ;

suppose A4: q `2 < 0 ; :: thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) )

then FanS (cn,q) = |.q.| * |[((((q `1) / |.q.|) - cn) / (1 - cn)),(- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| by A3, Def8
.= |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| by EUCLID:58 ;
hence ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) by A4, Def9, Th114; :: thesis: verum

end;

suppose A5: q `2 >= 0 ; :: thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) )

then A6: (cn -FanMorphS) . q = q by Th113;
A7: |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) by JGRAPH_3:1;
A8: 1 - cn > 0 by A2, XREAL_1:149;
A9: q `2 = 0 by A3, A5;
|.q.| <> 0 by A3, TOPRNS_1:24;
then |.q.| ^2 > 0 by SQUARE_1:12;
then ((q `1) ^2) / (|.q.| ^2) = 1 ^2 by A7, A9, XCMPLX_1:60;
then ((q `1) / |.q.|) ^2 = 1 ^2 by XCMPLX_1:76;
then A10: sqrt (((q `1) / |.q.|) ^2) = 1 ;

A11: now :: thesis: not q `1 < 0

assume q `1 < 0 ; :: thesis: contradiction
then - ((q `1) / |.q.|) = 1 by A10, SQUARE_1:23;
hence contradiction by A1, A3; :: thesis: verum

end;

sqrt (|.q.| ^2) = |.q.| by SQUARE_1:22;
then A12: |.q.| = q `1 by A7, A9, A11, SQUARE_1:22;
then 1 = (q `1) / |.q.| by A3, TOPRNS_1:24, XCMPLX_1:60;
then (((q `1) / |.q.|) - cn) / (1 - cn) = 1 by A8, XCMPLX_1:60;
hence ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) by A2, A6, A9, A12, EUCLID:53, TOPRNS_1:24, XCMPLX_1:60; :: thesis: verum

end;

end;

end;

suppose A13: ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ; :: thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) )

per cases ( q `2 < 0 or q `2 >= 0 ) ;

suppose q `2 < 0 ; :: thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) )

hence ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) by Th113, Th114; :: thesis: verum

end;

suppose A14: q `2 >= 0 ; :: thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) )

then A15: q `2 = 0 by A13;
A16: 1 + cn > 0 by A1, XREAL_1:148;
A17: |.q.| <> 0 by A13, TOPRNS_1:24;
1 > (q `1) / |.q.| by A2, A13, XXREAL_0:2;
then 1 * |.q.| > ((q `1) / |.q.|) * |.q.| by A17, XREAL_1:68;
then A18: ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & |.q.| > q `1 ) by A13, JGRAPH_3:1, TOPRNS_1:24, XCMPLX_1:87;
then A19: |.q.| = - (q `1) by A15, SQUARE_1:40;
A20: q `1 = - |.q.| by A15, A18, SQUARE_1:40;
then - 1 = (q `1) / |.q.| by A13, TOPRNS_1:24, XCMPLX_1:197;
then (((q `1) / |.q.|) - cn) / (1 + cn) = (- (1 + cn)) / (1 + cn)
.= - 1 by A16, XCMPLX_1:197 ;
then |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| = q by A15, A19, EUCLID:53;
hence ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) by A1, A14, A17, A20, Th113, XCMPLX_1:197; :: thesis: verum

end;

end;

end;

suppose ( q `2 > 0 or q = 0. (TOP-REAL 2) ) ; :: thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) )

hence ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) ; :: thesis: verum

end;

end;