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

theorem Th18:
  for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 &
  q`2>0 holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds p`2>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;
  now
    per cases;
    case
      q`1/|.q.|>=cn;
      hence thesis by A2,A3,JGRAPH_4:75;
    end;
    case
      q`1/|.q.|<cn;
      hence thesis by A1,A3,JGRAPH_4:76;
    end;
  end;
  hence thesis;
end;
