
theorem Th1:
  for p being Point of TOP-REAL 2 st |.p.|<=1 holds -1<=p`1 & p`1<=
  1 & -1<=p`2 & p`2<=1
proof
  let p be Point of TOP-REAL 2;
  set a=|.p.|;
A1: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
  then a^2-(p`1)^2>=0 by XREAL_1:63;
  then a^2-(p`1)^2+(p`1)^2>=0+(p`1)^2 by XREAL_1:7;
  then
A2: -a<=p`1 & p`1<=a by SQUARE_1:47;
  a^2-(p`2)^2>=0 by A1,XREAL_1:63;
  then a^2-(p`2)^2+(p`2)^2>=0+(p`2)^2 by XREAL_1:7;
  then
A3: -a<=p`2 & p`2<=a by SQUARE_1:47;
  assume
A4: |.p.|<=1;
  then -a>=-1 by XREAL_1:24;
  hence thesis by A4,A2,A3,XXREAL_0:2;
end;
