reserve A,B,C for Ordinal,
        o for object,
        x,y,z,t,r,l for Surreal,
        X,Y for set;

theorem
  0_No < z & x*z < y*z implies x < y
proof
  assume that A1: 0_No < z and A2:x*z < y*z;
  assume y <=x;
  then per cases;
  suppose y==x;
    then y*z==x*z by Th51;
    hence thesis by A2;
  end;
  suppose y < x;
    then y*z <= x*z by A1,Th70;
    hence thesis by A2;
  end;
end;
