reserve x, y, z, r, s, t for Real;

theorem
  |.x+y.|/(1 + |.x+y.|) <= |.x.|/(1 + |.x.|) + |.y.|/(1 + |.y.|)
proof
A1: s <= t & 0 < 1 + s & 0 < 1 + t implies s/(1 + s) <= t/(1 + t)
  proof
    assume that
A2: s <= t and
A3: 0 < 1 + s and
A4: 0 < 1 + t;
    s * 1 + s * t <= t + s * t by A2,XREAL_1:6;
    then s * (1 + t) * (1 + s)" <= t * (1 + s) * (1 + s)" by A3,XREAL_1:64;
    then s * (1 + t) * (1 + s)" <= t * ((1 + s) * (1 + s)");
    then s * (1 + t) * (1 + s)" <= t * 1 by A3,XCMPLX_0:def 7;
    then (s * (1 + s)") * (1 + t) * (1 + t)" <= t * (1 + t)" by A4,XREAL_1:64;
    then (s * (1 + s)") * ((1 + t) * (1 + t)") <= t * (1 + t)";
    then (s * (1 + s)") * 1 <= t * (1 + t)" by A4,XCMPLX_0:def 7;
    then s/(1 + s) <= t * (1 + t)" by XCMPLX_0:def 9;
    hence thesis by XCMPLX_0:def 9;
  end;
  set a = |.x.|, b = |.y.|, c = |.x+y.|;
A5: 0 <= a by COMPLEX1:46;
A6: 0 <= b by COMPLEX1:46;
A7: 0 + 0 < 1 + a by COMPLEX1:46,XREAL_1:8;
A8: 0 < 1 + a & 0 < 1 + a + b implies a/(1 + a + b) <= a/(1 + a)
  proof
    assume that
A9: 0 < 1 + a and
A10: 0 < 1 + a + b;
    0 + a <= a * b + a by A5,A6,XREAL_1:6;
    then a * 1 + a * a <= a * (1 + b) + a * a by XREAL_1:6;
    then a * (1 + a) * (1 + a)" <= a * (1 + a + b) * (1 + a)" by A9,XREAL_1:64;
    then a * ((1 + a) * (1 + a)") <= a * (1 + a + b) * (1 + a)";
    then a * 1 <= a * (1 + a + b) * (1 + a)" by A7,XCMPLX_0:def 7;
    then a * (1 + a + b)" <= (a * (1 + a)") * (1 + a + b) * (1 + a + b)" by A10
,XREAL_1:64;
    then a * (1 + a + b)" <= (a * (1 + a)") * ((1 + a + b) * (1 + a + b)");
    then a * (1 + a + b)" <= (a * (1 + a)") * 1 by A5,A6,XCMPLX_0:def 7;
    then a/(1 + a + b) <= a * (1 + a)" by XCMPLX_0:def 9;
    hence thesis by XCMPLX_0:def 9;
  end;
A11: 0 + 0 < 1 + b by COMPLEX1:46,XREAL_1:8;
A12: 0 < 1 + b & 0 < 1 + a + b implies b/(1 + a + b) <= b/(1 + b)
  proof
    assume that
A13: 0 < 1 + b and
A14: 0 < 1 + a + b;
    0 + b <= a * b + b by A5,A6,XREAL_1:6;
    then b * 1 + b * b <= (1 + a) * b + b * b by XREAL_1:6;
    then (b * (1 + b)) * (1 + b)" <= (b * (1 + a + b)) * (1 + b)" by A13,
XREAL_1:64;
    then b * ((1 + b) * (1 + b)") <= (b * (1 + a + b)) * (1 + b)";
    then b * 1 <= (b * (1 + a + b)) * (1 + b)" by A11,XCMPLX_0:def 7;
    then
    b * (1 + a + b)" <= ((b * (1 + b)") * (1 + a + b )) * (1 + a + b)" by A14,
XREAL_1:64;
    then b * (1 + a + b)" <= (b * (1 + b)") * ((1 + a + b) *(1 + a + b)");
    then b * (1 + a + b)" <= (b * (1 + b)") * 1 by A5,A6,XCMPLX_0:def 7;
    then b/(1 + a + b) <= b * (1 + b)" by XCMPLX_0:def 9;
    hence thesis by XCMPLX_0:def 9;
  end;
  0 + 0 < 1 + c by COMPLEX1:46,XREAL_1:8;
  then
A15: c/(1 + c) <= (a + b)/(1 + (a + b)) by A5,A6,A1,COMPLEX1:56;
  (a + b)/(1 + a + b) = a/(1 + a + b) + b/(1 + a + b) by XCMPLX_1:62;
  then (a + b)/(1 + a + b) <= a/(1 + a) + b/(1 + b) by A6,A7,A8,A12,XREAL_1:7;
  hence thesis by A15,XXREAL_0:2;
end;
