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

theorem Th22:
  for sn being Real,q being Point of TOP-REAL 2 st -1<sn & sn<1 &
  q`1>0 holds for p being Point of TOP-REAL 2 st p=(sn-FanMorphE).q holds p`1>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;
  now
    per cases;
    case
      q`2/|.q.|>=sn;
      hence thesis by A2,A3,JGRAPH_4:106;
    end;
    case
      q`2/|.q.|<sn;
      hence thesis by A1,A3,JGRAPH_4:107;
    end;
  end;
  hence thesis;
end;
