reserve a for Real;
reserve p,q for Point of TOP-REAL 2;

theorem
  for sn being Real,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
proof
  let sn be Real,q be Point of TOP-REAL 2;
  assume that
A1: q`1<0 and
A2: q`2/|.q.|=sn;
  let p be Point of TOP-REAL 2;
A3: |.q.|>0 by A1,Lm1,JGRAPH_2:3;
  assume p=(sn-FanMorphW).q;
  then
A4: p=|[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/
  (1-sn))]| by A1,A2,Th16;
  then p`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
  hence thesis by A2,A4,A3,EUCLID:52,XREAL_1:132;
end;
