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