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

theorem Th50:
  for cn being Real holds (q`1/|.q.|<=cn & q`2>0 implies cn
-FanMorphN.q= |[ |.q.|*((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/
  (1+cn))^2))]|)
proof
  let cn be Real;
  assume that
A1: q`1/|.q.|<=cn and
A2: q`2>0;
  per cases by A1,XXREAL_0:1;
  suppose
    q`1/|.q.|<cn;
    then
    FanN(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1+cn), sqrt(1-((q`1/|.q.|-cn)/(1+
    cn))^2)]| by A2,Def4
      .= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(
    1+cn))^2))]| by EUCLID:58;
    hence thesis by Def5;
  end;
  suppose
A3: q`1/|.q.|=cn;
    then (q`1/|.q.|-cn)/(1-cn)=0;
    hence thesis by A2,A3,Th49;
  end;
end;
