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.|/(1+|.x+y.|)<=|.x.|/(1+|.x.|)+|.y.|/(1+|.y.|)
proof
  per cases;
  suppose
    |.x+y.|=0;
    hence thesis;
  end;
  suppose
A1: |.x+y.|>0;
    1+|.y.|>=1 by XREAL_1:31;
    then 1+|.y.|+|.x.|>=1+|.x.| by XREAL_1:6;
    then
A2: |.x.|/(1+|.x.|+|.y.|)<=|.x.|/(1+|.x.|) by XREAL_1:118;
    1+|.x.|>=1 by XREAL_1:31;
    then 1+|.x.|+|.y.|>=1+|.y.| by XREAL_1:6;
    then |.y.|/(1+|.x.|+|.y.|)<=|.y.|/(1+|.y.|) by XREAL_1:118;
    then
A3: |.x.|/(1+|.x.|+|.y.|)+|.y.|/(1+|.x.|+|.y.|)<= |.x.|/(1+|.x.|)
    +|.y.|/(1+|.y.|) by A2,XREAL_1:7;
    1/(|.x.|+|.y.|)<=1/|.x+y.| by A1,COMPLEX1:56,XREAL_1:118;
    then 1/(|.x.|+|.y.|)+1<=1/|.x+y.|+1 by XREAL_1:7;
    then
A4: 1/(1/|.x+y.|+1)<=1/(1/(|.x.|+|.y.|)+1) by XREAL_1:118;
A5: 0<|.x.|+|.y.| by A1,COMPLEX1:56;
    then
A6: 1/(1/(|.x.|+|.y.|)+1) =(1*(|.x.|+|.y.|))/((1/(|.x.|+|.y.|)+1)*(
    |.x.|+|.y.|)) by XCMPLX_1:91
      .=(|.x.|+|.y.|)/(1/(|.x.|+|.y.|)*(|.x.|+|.y.|)+(|.x.|+|.y.|))
      .=(|.x.|+|.y.|)/(1+(|.x.|+|.y.|)) by A5,XCMPLX_1:87
      .=|.x.|/(1+|.x.|+|.y.|)+|.y.|/(1+|.x.|+|.y.|) by XCMPLX_1:62;
    |.x+y.|/(1+|.x+y.|) =(|.x+y.|/|.x+y.|)/((1+|.x+y.|)/|.x+y.|) by A1,
XCMPLX_1:55
      .=1/((1+|.x+y.|)/|.x+y.|) by A1,XCMPLX_1:60
      .=1/(1/|.x+y.|+|.x+y.|/|.x+y.|) by XCMPLX_1:62
      .=1/(1/|.x+y.|+1) by A1,XCMPLX_1:60;
    hence thesis by A4,A6,A3,XXREAL_0:2;
  end;
end;
