reserve a,b,c,x,y,z for Real;

theorem
  0<=x & x<=1 implies x^2<=x
proof
  assume that
A1: 0<=x and
A2: x<=1;
  per cases by A1;
  suppose
    0=x;
    hence thesis;
  end;
  suppose
A3: 0<x;
    per cases by A2,XXREAL_0:1;
    suppose
      x=1;
      hence thesis;
    end;
    suppose
      x<1;
      hence thesis by A3,Th13;
    end;
  end;
end;
