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

theorem Th82:
  for sn being Real holds (q`2/|.q.|>=sn & q`1>0 implies sn
-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-
  sn)/(1-sn))]|)& (q`1<=0 implies sn-FanMorphE.q=q)
proof
  let sn be Real;
  hereby
    assume q`2/|.q.|>=sn & q`1>0;
    then
    FanE(sn,q)= |.q.|*|[sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2), (q`2/|.q.|-sn)/
    (1-sn)]| by Def6
      .= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn
    )/(1-sn))]| by EUCLID:58;
    hence sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|*
    ((q`2/|.q.|-sn)/(1-sn))]| by Def7;
  end;
  assume
A1: q`1<=0;
  sn-FanMorphE.q=FanE(sn,q) by Def7;
  hence thesis by A1,Def6;
end;
