let q be Point of (TOP-REAL 2); :: thesis: for cn being Real holds
( ( (q `1) / |.q.| >= cn & q `2 < 0 implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( q `2 >= 0 implies (cn -FanMorphS) . q = q ) )
let cn be Real; :: thesis: ( ( (q `1) / |.q.| >= cn & q `2 < 0 implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( q `2 >= 0 implies (cn -FanMorphS) . q = q ) )
hereby :: thesis: ( q `2 >= 0 implies (cn -FanMorphS) . q = q )
assume ( (q `1) / |.q.| >= cn & q `2 < 0 ) ; :: thesis: (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 Def8
.= |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| by EUCLID:58 ;
hence (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| by Def9; :: thesis: verum
end;
assume A1: q `2 >= 0 ; :: thesis: (cn -FanMorphS) . q = q
(cn -FanMorphS) . q = FanS (cn,q) by Def9;
hence (cn -FanMorphS) . q = q by A1, Def8; :: thesis: verum