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

theorem Th16:
  for sn being Real holds (q`2/|.q.|>=sn & q`1<0 implies sn
-FanMorphW.q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|
  -sn)/(1-sn))]|) & (q`1>=0 implies sn-FanMorphW.q=q)
proof
  let sn be Real;
  hereby
    assume q`2/|.q.|>=sn & q`1<0;
    then
    FanW(sn,q)= |.q.|*|[-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2), (q`2/|.q.|-sn)
    /(1-sn)]| by Def2
      .= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-
    sn)/(1-sn))]| by EUCLID:58;
    hence
    sn-FanMorphW.q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|*
    ((q`2/|.q.|-sn)/(1-sn))]| by Def3;
  end;
  assume
A1: q`1>=0;
  sn-FanMorphW.q=FanW(sn,q) by Def3;
  hence thesis by A1,Def2;
end;
