reserve x,y,t for Real;

theorem Th25:
  1<=y implies 0<y-sqrt(y^2-1)
proof
  assume
A1: 1<=y;
  then 1*y<=y*y by XREAL_1:64;
  then 1<=y^2 by A1,XXREAL_0:2;
  then
A2: 1-1<=y^2-1 by XREAL_1:13;
  -1+y^2<0+y^2 by XREAL_1:8;
  then sqrt(y^2-1)<sqrt(y^2) by A2,SQUARE_1:27;
  then sqrt(y^2-1)<y by A1,SQUARE_1:22;
  then sqrt(y^2-1)-sqrt(y^2-1)<y-sqrt(y^2-1) by XREAL_1:14;
  hence thesis;
end;
