let sn be Real; :: thesis: for q being Point of (TOP-REAL 2) st q `1 < 0 & (q `2 ) / |.q.| = sn holds
for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW ) . q holds
( p `1 < 0 & p `2 = 0 )
let q be Point of (TOP-REAL 2); :: thesis: ( q `1 < 0 & (q `2 ) / |.q.| = sn implies for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW ) . q holds
( p `1 < 0 & p `2 = 0 ) )
assume that
A1: q `1 < 0 and
A2: (q `2 ) / |.q.| = sn ; :: thesis: for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW ) . q holds
( p `1 < 0 & p `2 = 0 )
let p be Point of (TOP-REAL 2); :: thesis: ( p = (sn -FanMorphW ) . q implies ( p `1 < 0 & p `2 = 0 ) )
A3: |.q.| > 0 by A1, Lm1, JGRAPH_2:11;
assume p = (sn -FanMorphW ) . q ; :: thesis: ( p `1 < 0 & p `2 = 0 )
then A4: p = |[(|.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 ))))),(|.q.| * ((((q `2 ) / |.q.|) - sn) / (1 - sn)))]| by A1, A2, Th23;
then p `1 = |.q.| * (- (sqrt (1 - (((((q `2 ) / |.q.|) - sn) / (1 - sn)) ^2 )))) by EUCLID:56;
hence ( p `1 < 0 & p `2 = 0 ) by A2, A4, A3, Lm13, EUCLID:56, XREAL_1:134; :: thesis: verum