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

theorem
  0_No <= x & y <= z implies y*x <= z*x
proof
  assume A1: 0_No <= x & y <= z;
  per cases;
  suppose x <= 0_No;
    then x == 0_No by A1;
    then y*x == y * 0_No = 0_No = z * 0_No == z*x by Th51;
     hence thesis by SURREALO:4;
  end;
  suppose A2: 0_No < x;
    per cases;
      suppose z <= y;
        then z==y by A1;
        then y*x == z*x by Th51;
        hence thesis;
      end;
     suppose y < z;
       hence thesis by A2,Th70;
      end;
    end;
end;
