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

theorem Th49:
  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))]|)& (q`2<=0 implies cn-FanMorphN.q=q)
proof
  let cn be Real;
  hereby
    assume q`1/|.q.|>=cn & q`2>0;
    then
    FanN(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1-cn), sqrt(1-((q`1/|.q.|-cn)/(1-
    cn))^2)]| by Def4
      .= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(sqrt(1-((q`1/|.q.|-cn)/(1
    -cn))^2))]| by EUCLID:58;
    hence cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(sqrt(1-((q
    `1/|.q.|-cn)/(1-cn))^2))]| by Def5;
  end;
  assume
A1: q`2<=0;
  cn-FanMorphN.q=FanN(cn,q) by Def5;
  hence thesis by A1,Def4;
end;
