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-FanMorphE).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;
A3: |.q.|<>0 & sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>0 by A1,A2,JGRAPH_2:3
,TOPRNS_1:24;
  let p be Point of TOP-REAL 2;
  assume p=(sn-FanMorphE).q;
  then p=|[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/(
  1-sn))]| by A1,A2,Th82;
  hence thesis by A2,A3,EUCLID:52;
end;
