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.|<1 & |.y.|<1 implies |.(x+y)/(1+x*y).|<=1
proof
  assume that
A1: |.x.|<1 and
A2: |.y.|<1;
  per cases;
  suppose
A3: 1+x*y<>0;
    y^2<1 by A2,Lm7;
    then
A4: y^2-y^2<1-y^2 by XREAL_1:14;
    x^2<1 by A1,Lm7;
    then x^2-x^2<1-x^2 by XREAL_1:14;
    then 0<(1-x^2)*(1-y^2) by A4;
    then 0+(x^2+y^2)<1-y^2-x^2+x^2*y^2+(x^2+y^2) by XREAL_1:8;
    then
A5: x^2+y^2+2*x*y<1+x^2*y^2+2*x*y by XREAL_1:8;
    (1+x*y)^2>0 by A3,SQUARE_1:12;
    then (x+y)^2/(1+x*y)^2<(x*y+1)^2/(1+x*y)^2 by A5,XREAL_1:74;
    then (x+y)^2/(1+x*y)^2<1 by A3,XCMPLX_1:60;
    then ((x+y)/(1+x*y))^2<1 by XCMPLX_1:76;
    hence thesis by Lm8;
  end;
  suppose
    1+x*y=0;
    then |.(x+y)/(1+x*y).|= |.0.| by XCMPLX_1:49
      .=0 by ABSVALUE:2;
    hence thesis;
  end;
end;
