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

theorem
  for sn being Real,q being Point of TOP-REAL 2 st -1<sn & sn<1 & q`1>=0
  & q`2/|.q.|<sn & |.q.|<>0 holds for p being Point of TOP-REAL 2 st p=(sn
  -FanMorphE).q holds p`1>=0 & p`2<0
proof
  let sn be Real,q be Point of TOP-REAL 2;
  assume that
A1: -1<sn and
A2: sn<1 and
A3: q`1>=0 & q`2/|.q.|<sn and
A4: |.q.|<>0;
  let p be Point of TOP-REAL 2;
  assume
A5: p=(sn-FanMorphE).q;
  now
    per cases;
    case
A6:   q`1=0;
      then |.q.|^2=(q`2)^2+0^2 by JGRAPH_3:1
        .=(q`2)^2;
      then
A7:   |.q.|=q`2 or |.q.|=-(q`2) by SQUARE_1:40;
      q=p by A5,A6,JGRAPH_4:82;
      hence thesis by A2,A3,A4,A7,XCMPLX_1:60;
    end;
    case
      q`1<>0;
      hence thesis by A1,A3,A5,JGRAPH_4:107;
    end;
  end;
  hence thesis;
end;
