reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem Th18:
  x infinitely< z & y infinitely< z implies x + y infinitely< z
proof
  assume
A1: x infinitely< z & y infinitely< z;
  let r be positive Real;
  set R=uReal.r,H=uReal.(1/2);
A2: 1/2+1/2=1;
  x infinitely< z * H & y infinitely< z * H by A1,Th13;
  then x * R < z *H & y * R <= z *H;
  then
A3:  x * R + y * R < z *H + z*H by SURREALR:44;
  z *H + z*H == z *(H + H) == z* 1_No =z
  by A2,SURREALN:55,48,SURREALR:51,SURREALR:67;
  then z *H + z*H == z by SURREALO:4;
  then (x + y) * R == x * R + y * R < z by A3,SURREALO:4,SURREALR:67;
  hence thesis by SURREALO:4;
end;
