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