FIRST SUPPLEMENTAL INDENTURE dated as of June 30, 2006

Exhibit 4.13(b)

FIRST SUPPLEMENTAL INDENTURE

dated as of June 30, 2006

among

LYONDELL CHEMICAL COMPANY,

as Company

the SUBSIDIARY GUARANTORS party hereto

and

THE BANK OF NEW YORK,

as Trustee

11 ¹/8% Senior Secured Notes due 2012

THIS FIRST SUPPLEMENTAL INDENTURE (this “Supplemental Indenture”), entered into as of June 30, 2006, among LYONDELL CHEMICAL COMPANY, a Delaware corporation (the “Company”), LRC HOLDINGS GP LLC, a Delaware limited liability company, LRC HOLDINGS LP LLC, a Delaware limited liability company, LRP HOLDINGS LP LLC, a Delaware limited liability company, LYONDELL (PELICAN) PETROCHEMICAL L.P.1, INC., a Delaware corporation, LYONDELL CHEMICAL DELAWARE COMPANY, a Delaware corporation, LYONDELL CHEMICAL PROPERTIES, L.P., a Delaware limited partnership, LYONDELL CHEMICAL TECHNOLOGY MANAGEMENT, INC., a Delaware corporation, LYONDELL CHIMIE FRANCE CORPORATION, a Delaware corporation, LYONDELL FRANCE, INC., a Delaware corporation, LYONDELL LP3 GP, LLC, a Delaware limited liability company, LYONDELL LP3 PARTNERS, LP, a Delaware limited partnership, LYONDELL LP4 INC., a Delaware corporation, LYONDELL PETROCHEMICAL L.P. INC., a Delaware corporation, LYONDELL REFINING COMPANY LP, a Delaware limited partnership, LYONDELL REFINING GP, LLC, a Delaware limited liability company, LYONDELL REFINING LP, LLC, a Delaware limited liability company, LYONDELL REFINING PARTNERS, LP, a Delaware limited partnership, POSM DELAWARE, INC., a Delaware corporation, POSM II PROPERTIES PARTNERSHIP, L.P., a Delaware limited partnership (each an “Undersigned”), and THE BANK OF NEW YORK, as trustee (the “Trustee”).

RECITALS

WHEREAS, the Company, the Subsidiary Guarantors party thereto and the Trustee entered into the Indenture, dated as of July 2, 2002 (the “Indenture”), relating to the Company’s 11 ¹/8% Senior Secured Notes due 2012 (the “Notes”);

WHEREAS, as a condition to the Trustee entering into the Indenture and the purchase of the Notes by the Holders, the Company agreed, subject to certain exceptions, pursuant to Section 4.22 of the Indenture to cause any Restricted Subsidiary that has guaranteed or secured Indebtedness of the Company or any of its Restricted Subsidiaries to provide Subsidiary Guarantees.

AGREEMENT

NOW, THEREFORE, in consideration of the premises and mutual covenants herein contained and intending to be legally bound, the parties hereto hereby agree as follows:

Section 1. Capitalized terms used herein and not otherwise defined herein are used as defined in the Indenture.

Section 2. Each Undersigned, by its execution of this Supplemental Indenture, agrees to be a Subsidiary Guarantor under the Indenture and to be bound by the terms of the Indenture applicable to Subsidiary Guarantors, including, but not limited to, Article 13 thereof.

Section 3. This Supplemental Indenture shall be governed by and construed in accordance with the internal laws of the State of New York.

Section 4. This Supplemental Indenture may be signed in various counterparts which together shall constitute one and the same instrument.

Section 5. This Supplemental Indenture is an amendment supplemental to the Indenture and said Indenture and this Supplemental Indenture shall henceforth be read together.

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IN WITNESS WHEREOF, the parties have duly executed and delivered this Supplemental Indenture or have caused this Supplemental Indenture to be duly executed on their respective behalf by their respective officers thereunto duly authorized, as of the day and year first above written.

Signature Page to Supplemental Indenture

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