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
  a/(a+b+d)+b/(a+b+c)+c/(b+c+d)+d/(a+c+d)>1
proof
  d+(a+b+c)>0+(a+b+c) by XREAL_1:8;
  then
A1: b/(a+b+c+d)<b/(a+b+c) by XREAL_1:76;
  (b+c+d)+a>0+(b+c+d) by XREAL_1:8;
  then
A2: c/(b+c+d)>c/(a+b+c+d) by XREAL_1:76;
  c+(a+b+d)>0+(a+b+d) by XREAL_1:8;
  then a/(a+b+c+d)<a/(a+b+d) by XREAL_1:76;
  then a/(a+b+c+d)+b/(a+b+c+d)<a/(a+b+d)+b/(a+b+c) by A1,XREAL_1:8;
  then (a+b)/(a+b+c+d)<a/(a+b+d)+b/(a+b+c) by XCMPLX_1:62;
  then (a+b)/(a+b+c+d)+c/(a+b+c+d)<(a/(a+b+d)+b/(a+b+c))+c/(b+c+d) by A2,
XREAL_1:8;
  then
A3: (a+b+c)/(a+b+c+d)<(a/(a+b+d)+b/(a+b+c))+c/(b+c+d) by XCMPLX_1:62;
  b+(a+c+d)>0+(a+c+d) by XREAL_1:8;
  then d/(a+c+d)>d/(a+b+c+d) by XREAL_1:76;
  then (a+b+c)/(a+b+c+d)+d/(a+b+c+d)<(a/(a+b+d)+b/(a+b+c))+c/(b+c+d)+d/(a+c+d
  ) by A3,XREAL_1:8;
  then (a+b+c+d)/(a+b+c+d)<(a/(a+b+d)+b/(a+b+c))+c/(b+c+d)+d/(a+c+d) by
XCMPLX_1:62;
  hence thesis by XCMPLX_1:60;
end;
