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

per cases ( (q `2) / |.q.| < sn or (q `2) / |.q.| = sn ) by A1, XXREAL_0:1;

suppose (q `2) / |.q.| < sn ; :: thesis: (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]|

then FanW (sn,q) = |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),((((q `2) / |.q.|) - sn) / (1 + sn))]| by A2, Def2
.= |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by EUCLID:58 ;
hence (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by Def3; :: thesis: verum

end;

suppose A3: (q `2) / |.q.| = sn ; :: thesis: (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]|

then (((q `2) / |.q.|) - sn) / (1 - sn) = 0 ;
hence (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by A2, A3, Th16; :: thesis: verum

end;

end;