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)<2
proof
A1: b/(a+b)>b/(a+b+c) by XREAL_1:29,76;
  a/(a+b)>a/(a+b+d) by XREAL_1:29,76;
  then a/(a+b)+b/(a+b)>a/(a+b+d)+b/(a+b+c) by A1,XREAL_1:8;
  then (a+b)/(a+b)>a/(a+b+d)+b/(a+b+c) by XCMPLX_1:62;
  then
A2: 1>a/(a+b+d)+b/(a+b+c) by XCMPLX_1:60;
  a+(c+d)>c+d by XREAL_1:29;
  then
A3: d/(c+d)>d/(a+c+d) by XREAL_1:76;
  b+(c+d)>c+d by XREAL_1:29;
  then c/(c+d)>c/(b+c+d) by XREAL_1:76;
  then 1+c/(c+d)>a/(a+b+d)+b/(a+b+c)+c/(b+c+d) by A2,XREAL_1:8;
  then (1+c/(c+d))+d/(c+d)>(a/(a+b+d)+b/(a+b+c)+c/(b+c+d))+d/(a+c+d) by A3,
XREAL_1:8;
  then 1+(c/(c+d)+d/(c+d))>a/(a+b+d)+b/(a+b+c)+c/(b+c+d)+d/(a+c+d);
  then 1+(c+d)/(c+d)>a/(a+b+d)+b/(a+b+c)+c/(b+c+d)+d/(a+c+d) by XCMPLX_1:62;
  then 1+1>a/(a+b+d)+b/(a+b+c)+c/(b+c+d)+d/(a+c+d) by XCMPLX_1:60;
  hence thesis;
end;
