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-FanMorphS).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;
  let p be Point of TOP-REAL 2;
A3: |.q.|<>0 by A1,JGRAPH_2:3,TOPRNS_1:24;
  assume p=(cn-FanMorphS).q;
  then
A4: p=|[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-
  cn))^2))]| by A1,A2,Th113;
  then p`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
  hence thesis by A2,A4,A3,EUCLID:52,XREAL_1:132;
end;
