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
  m*x+z*y<=sqrt(m^2+z^2)*sqrt(x^2+y^2)
proof
A1: z^2>=0 by XREAL_1:63;
A2: m^2>=0 by XREAL_1:63;
  then
A3: sqrt(m^2+z^2)>=0 by A1,SQUARE_1:def 2;
A4: (m*x+z*y)^2>=0 by XREAL_1:63;
  (m*y)^2+(z*x)^2>=2*(m*y)*(z*x) by SERIES_3:6;
  then
A5: (m*y)^2+(z*x)^2+(m^2*x^2+z^2*y^2)>=2*m*x*y*z+(m^2*x^2+z^2*y^2) by XREAL_1:6
;
A6: y^2>=0 by XREAL_1:63;
A7: x^2>=0 by XREAL_1:63;
  then
A8: sqrt(x^2+y^2)>=0 by A6,SQUARE_1:def 2;
  (sqrt(m^2+z^2)*sqrt(x^2+y^2))^2 =(sqrt(m^2+z^2))^2*(sqrt(x^2+y^2))^2
    .=(m^2+z^2)*(sqrt(x^2+y^2))^2 by A2,A1,SQUARE_1:def 2
    .=(m^2+z^2)*(x^2+y^2) by A7,A6,SQUARE_1:def 2
    .=m^2*x^2+(m*y)^2+(z*x)^2+z^2*y^2;
  then sqrt((sqrt(m^2+z^2)*sqrt(x^2+y^2))^2)>=sqrt((m*x+z*y)^2) by A5,A4,
SQUARE_1:26;
  then
A9: sqrt(m^2+z^2)*sqrt(x^2+y^2)>=sqrt((m*x+z*y)^2) by A3,A8,SQUARE_1:22;
  per cases;
  suppose
    m*x+z*y>0;
    hence thesis by A9,SQUARE_1:22;
  end;
  suppose
    m*x+z*y=0;
    hence thesis by A3,A8;
  end;
  suppose
    m*x+z*y<0;
    hence thesis by A3,A8;
  end;
end;
