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

theorem Th113:
  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))]|)& (q`2>=0 implies cn-FanMorphS.q=q)
proof
  let cn be Real;
  hereby
    assume q`1/|.q.|>=cn & q`2<0;
    then
    FanS(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1-cn), -sqrt(1-((q`1/|.q.|-cn)/(1-
    cn))^2)]| by Def8
      .= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(-sqrt(1-((q`1/|.q.|-cn)/(
    1-cn))^2))]| by EUCLID:58;
    hence
    cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(-sqrt(1-((q
    `1/|.q.|-cn)/(1-cn))^2))]| by Def9;
  end;
  assume
A1: q`2>=0;
  cn-FanMorphS.q=FanS(cn,q) by Def9;
  hence thesis by A1,Def8;
end;
