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

theorem Th57:
  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-FanMorphS).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;
  let p be Point of TOP-REAL 2;
  assume
A4: p=(cn-FanMorphS).q;
  per cases by A3;
  suppose
A5: q`2<0;
    now
      per cases;
      case
        q`1/|.q.|<cn;
        hence thesis by A1,A4,A5,JGRAPH_4:138;
      end;
      case
        q`1/|.q.|>=cn;
        hence thesis by A2,A4,A5,JGRAPH_4:137;
      end;
    end;
    hence thesis;
  end;
  suppose
    q`2=0;
    hence thesis by A4,JGRAPH_4:113;
  end;
end;
