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

theorem
  for cn being Real,q being Point of TOP-REAL 2 st q`2>0 & q`1/|.q.|=cn
holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds p`2>0 & p`1=0
proof
  let cn be Real,q be Point of TOP-REAL 2;
  assume that
A1: q`2>0 and
A2: q`1/|.q.|=cn;
A3: |.q.|<>0 & sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>0 by A1,A2,JGRAPH_2:3
,TOPRNS_1:24;
  let p be Point of TOP-REAL 2;
  assume p=(cn-FanMorphN).q;
  then p=|[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-
  cn))^2))]| by A1,A2,Th49;
  hence thesis by A2,A3,EUCLID:52;
end;
