reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem Th9:
  [.-sqrt 2/2,sqrt 2/2.] c= ].-1,1.[
proof
A1: sqrt 2 > 0 by SQUARE_1:25;
  sqrt 2 < sqrt 4 by SQUARE_1:27;
  then
A2: sqrt 2/2 < 2/2 by SQUARE_1:20,XREAL_1:74;
  then sqrt 2/2*(-1) > 1*(-1) by XREAL_1:69;
  then
A3: -sqrt 2/2 in ].-1,1.[ by A1,XXREAL_1:4;
  sqrt 2/2 > 0 by A1,XREAL_1:139;
  then sqrt 2/2 in ].-1,1.[ by A2,XXREAL_1:4;
  hence thesis by A3,XXREAL_2:def 12;
end;
