reserve x,y,t for Real;

theorem Th22:
  0<x & x<=1 implies 1-x^2>=0
proof
  assume that
A1: 0<x and
A2: x<=1;
A3: x-1<=1-1 by A2,XREAL_1:9;
  per cases by A3;
  suppose
    x-1<0;
    then x-1+1<0+1 by XREAL_1:8;
    hence thesis by A1,Lm5;
  end;
  suppose
    x-1=0;
    hence thesis;
  end;
end;
