reserve a,b,c,d for positive Real,
  m,u,w,x,y,z for Real,
  n,k for Nat,
  s,s1 for Real_Sequence;

theorem
  x+y<=sqrt(2*(x^2+y^2))
proof
  x^2+y^2>=2*x*y by SERIES_3:6;
  then
A1: x^2+y^2+(x^2+y^2)>=2*x*y+(x^2+y^2) by XREAL_1:6;
  then
A2: 2*(x^2+y^2)>=(x+y)^2;
A3: (x+y)^2>=0 by XREAL_1:63;
  then
A4: sqrt(2*(x^2+y^2))>=sqrt((x+y)^2) by A1,SQUARE_1:26;
  per cases;
  suppose
    x+y>0;
    hence thesis by A4,SQUARE_1:22;
  end;
  suppose
    x+y=0;
    hence thesis by A2,SQUARE_1:17,26;
  end;
  suppose
    x+y<0;
    hence thesis by A1,A3,SQUARE_1:def 2;
  end;
end;
