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

theorem  Th45:
  |.y.| infinitely< |.x.| implies not x + y == 0_No
proof
  assume
A1: |.y.| infinitely< |.x.|;
  per cases by Def6;
  suppose |.x.| = x;
    hence thesis by A1,Th44;
  end;
  suppose |.x.| = -x;
    then
A2: |.-y.| infinitely< -x by Th40,Th17,A1;
    -x + -y = - (x+y) by SURREALR:40;
    hence thesis by A2,Th44,SURREALR:24;
  end;
end;
