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

theorem  Th44:
  |.y.| infinitely< x implies not x + y == 0_No
proof
  assume
A1: |.y.| infinitely< x;
  then
A2: |.y.| < x by Th9;
  per cases;
  suppose
A3: 0_No <= y;
    then y < x by A2,Def6;
    then
A4: y+y < x+y by SURREALR:44;
    0_No = 0_No+0_No <= y+y by A3,SURREALR:43;
    hence thesis by A4,SURREALO:4;
  end;
  suppose y < 0_No;
    then |.y.| = - y by Def6;
    then
A5: -y+y < x+y by A1,Th9,SURREALR:44;
    y-y == 0_No by SURREALR:39;
    hence thesis by A5,SURREALO:4;
  end;
end;
