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

theorem Th114:
  for cn being Real holds (q`1/|.q.|<=cn & q`2<0 implies cn
-FanMorphS.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
    FanS(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1+cn), -sqrt(1-((q`1/|.q.|-cn)/(1+
    cn))^2)]| by A2,Def8
      .= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/
    (1+cn))^2))]| by EUCLID:58;
    hence thesis by Def9;
  end;
  suppose
A3: q`1/|.q.|=cn;
    then (q`1/|.q.|-cn)/(1-cn)=0;
    hence thesis by A2,A3,Th113;
  end;
end;
