let q be Point of (TOP-REAL 2); :: thesis: for sn being real number holds
( ( (q `2 ) / |.q.| >= sn & q `1 > 0 implies (sn -FanMorphE ) . q = |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| ) & ( q `1 <= 0 implies (sn -FanMorphE ) . q = q ) )
let sn be real number ; :: thesis: ( ( (q `2 ) / |.q.| >= sn & q `1 > 0 implies (sn -FanMorphE ) . q = |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| ) & ( q `1 <= 0 implies (sn -FanMorphE ) . q = q ) )
A1: (sn -FanMorphE ) . q = FanE sn,q by Def7;

hereby :: thesis: ( q `1 <= 0 implies (sn -FanMorphE ) . q = q )

assume ( (q `2 ) / |.q.| >= sn & q `1 > 0 ) ; :: thesis: (sn -FanMorphE ) . q = |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]|
then FanE sn,q = |.q.| * |[(sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))),((((q `2 ) / |.q.|) - sn) / (1 - sn))]| by Def6
.= |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by EUCLID:62 ;
hence (sn -FanMorphE ) . q = |[(|.q.| * (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by Def7; :: thesis: verum

end;

assume q `1 <= 0 ; :: thesis: (sn -FanMorphE ) . q = q
hence (sn -FanMorphE ) . q = q by A1, Def6; :: thesis: verum