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

theorem Th17:
  for sn being Real st q`2/|.q.|<=sn & q`1<0 holds sn-FanMorphW.q=
|[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|*((q`2/|.q.|-sn)/(1+sn))]|
proof
  let sn be Real;
  assume that
A1: q`2/|.q.|<=sn and
A2: q`1<0;
  per cases by A1,XXREAL_0:1;
  suppose
    q`2/|.q.|<sn;
    then
    FanW(sn,q)= |.q.|*|[-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2), (q`2/|.q.|-sn)
    /(1+sn)]| by A2,Def2
      .= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-
    sn)/(1+sn))]| by EUCLID:58;
    hence thesis by Def3;
  end;
  suppose
A3: q`2/|.q.|=sn;
    then (q`2/|.q.|-sn)/(1-sn)=0;
    hence thesis by A2,A3,Th16;
  end;
end;
