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

theorem Th41:
  |.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:51,67;
  then z *H + z*H == z by SURREALO:4;
  then
A4: |. x .|* R + |. y .|* R < z by A3,SURREALO:4;
A5: 0_No <= R by SURREALI:def 8;
  |.x + y.| * R <= (|.x.| + |.y.|) * R ==
  |. x .|* R + |. y .|* R by Th37,A5,SURREALR:75,67;
  then |.x + y.| * R <= |. x .|* R + |. y .|* R by SURREALO:4;
  hence thesis by A4,SURREALO:4;
end;
