
theorem Th2:
  for p being Point of TOP-REAL 2 st |.p.|<=1 & p`1<>0 & p`2<>0
  holds -1<p`1 & p`1<1 & -1<p`2 & p`2<1
proof
  let p be Point of TOP-REAL 2;
  assume that
A1: |.p.|<=1 and
A2: p`1<>0 and
A3: p`2<>0;
  set a=|.p.|;
A4: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
  then a^2-(p`1)^2+(p`1)^2>0+(p`1)^2 by A3,SQUARE_1:12,XREAL_1:8;
  then
A5: -a<p`1 & p`1<a by SQUARE_1:48;
  a^2-(p`2)^2+(p`2)^2>0+(p`2)^2 by A2,A4,SQUARE_1:12,XREAL_1:8;
  then
A6: -a<p`2 & p`2<a by SQUARE_1:48;
  -a>=-1 by A1,XREAL_1:24;
  hence thesis by A1,A5,A6,XXREAL_0:2;
end;
