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

theorem Th83:
  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))]|)
proof
  let sn be Real;
  assume that
A1: q`2/|.q.|<=sn and
A2: q`1>0;
  now
    per cases by A1,XXREAL_0:1;
    case
      q`2/|.q.|<sn;
      then
      FanE(sn,q)= |.q.|*|[sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2), (q`2/|.q.|-sn
      )/(1+sn)]| by A2,Def6
        .= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-
      sn)/(1+sn))]| by EUCLID:58;
      hence thesis by Def7;
    end;
    case
A3:   q`2/|.q.|=sn;
      then (q`2/|.q.|-sn)/(1-sn)=0;
      hence thesis by A2,A3,Th82;
    end;
  end;
  hence thesis;
end;
