 reserve A,B,O for Ordinal,
      n,m for Nat,
      a,b,o for object,
      x,y,z for Surreal,
      X,Y,Z for set,
      Inv,I1,I2 for Function;

theorem
  not x==0_No & not y == 0_No implies (x*y)" == x" * y"
proof
  assume
A1: not x==0_No & not y == 0_No; then
A2: x*x" == 1_No == y*y" by Th33;
  (x" * y") * (x*y) == (x" * y") * y*x == (x" * (y" * y))*x by SURREALR:54,69;
  then (x" * y") * (x*y) == (y" * y)*x"*x == (y" * y)*(x"*x)
  by SURREALO:4,SURREALR:69;
  then (x" * y") * (x*y) == (y" * y)*(x"*x) == (y" * y)*1_No
  by A2,SURREALO:4,SURREALR:54;
  then (x" * y") * (x*y) == y" * y by SURREALO:4;
  then (x" * y") * (x*y) == 1_No by A2,SURREALO:4;
  hence thesis by A1,Lm3,Th41;
end;
