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

theorem Th74:
  x*y < 0_No iff (x < 0_No & 0_No < y ) or (0_No < x & y < 0_No)
proof
  set z=-y;
  thus x*y < 0_No implies (x < 0_No & 0_No < y ) or (0_No < x & y < 0_No)
  proof
    assume x*y < 0_No;
    then 0_No < -(x*y)=x*z by Th23,Th10,Th58;
    then (x < 0_No & z < 0_No) or (0_No < x & 0_No < z) by Th72;
    hence thesis by Th23,Th10;
  end;
  assume (x < 0_No & 0_No < y ) or (0_No < x & y < 0_No);
  then (x < 0_No & z < 0_No) or (0_No < x & 0_No < z) by Th23,Th10;
  then 0_No < x*z = - (x*y) by Th72,Th58;
  hence thesis by Th23,Th10;
end;
