reserve x,y,t for Real;

theorem
  0<x & x<1 implies 1/(sqrt(1-x^2))>1
proof
  assume that
A1: 0<x and
A2: x<1;
A3: 0<1-x^2 by A1,A2,Lm5;
  then
A4: 0<sqrt(1-x^2) by SQUARE_1:25;
  x*x>0*x by A1;
  then x^2*(-1)<0*(-1);
  then -x^2+1<0+1 by XREAL_1:8;
  then sqrt(1-x^2)<1 by A3,SQUARE_1:18,27;
  hence thesis by A4,XREAL_1:187;
end;
