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 Th29:
  (m*x+u*y+w*z)^2<=(m^2+u^2+w^2)*(x^2+y^2+z^2)
proof
  m^2*z^2+w^2*x^2=(m*z)^2+(w*x)^2;
  then
A1: m^2*z^2+w^2*x^2>=2*(m*z)*(w*x) by SERIES_3:6;
  u^2*z^2+w^2*y^2=(u*z)^2+(w*y)^2;
  then
A2: u^2*z^2+w^2*y^2>=2*(u*z)*(w*y) by SERIES_3:6;
  (m*y)^2+(u*x)^2>=2*(m*y)*(u*x) by SERIES_3:6;
  then m^2*y^2+u^2*x^2+(m^2*z^2+w^2*x^2)>=2*m*y*u*x+2*m*z*w*x by A1,XREAL_1:7;
  then
  m^2*y^2+u^2*x^2+m^2*z^2+w^2*x^2+(u^2*z^2+w^2*y^2)>=2*m*y*u*x+2*m*z*w *x+
  2*u*z*w*y by A2,XREAL_1:7;
  then
  m^2*y^2+u^2*x^2+m^2*z^2+w^2*x^2+(u^2*z^2)+(w^2*y^2)+((m*x)^2+(u*y)^2+ (w
  *z)^2)>=2*m*y*u*x+2*m*z*w*x+2*u*z*w*y+((m*x)^2+(u*y)^2+(w*z)^2) by XREAL_1:6;
  hence thesis;
end;
