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

theorem Th20:
  for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 &
q`2>=0 & q`1/|.q.|<cn & |.q.|<>0 holds for p being Point of TOP-REAL 2 st p=(cn
  -FanMorphN).q holds p`2>=0 & p`1<0
proof
  let cn be Real,q be Point of TOP-REAL 2;
  assume that
A1: -1<cn and
A2: cn<1 and
A3: q`2>=0 & q`1/|.q.|<cn and
A4: |.q.|<>0;
  let p be Point of TOP-REAL 2;
  assume
A5: p=(cn-FanMorphN).q;
  now
    per cases;
    case
A6:   q`2=0;
      then |.q.|^2=(q`1)^2+0^2 by JGRAPH_3:1
        .=(q`1)^2;
      then
A7:   |.q.|=q`1 or |.q.|=-(q`1) by SQUARE_1:40;
      q=p by A5,A6,JGRAPH_4:49;
      hence thesis by A2,A3,A4,A7,XCMPLX_1:60;
    end;
    case
      q`2<>0;
      hence thesis by A1,A3,A5,JGRAPH_4:76;
    end;
  end;
  hence thesis;
end;
