A method of producing oxygen product comprising: separating air by a cryogenic rectification process employing a plurality of air separation units (3, 4) having higher pressure columns (44, 46) and a lower pressure columns (54, 56) operatively associated with the higher pressure columns (44, 46) to produce oxygen-rich streams (80, 82) that are utilized in producing the oxygen product, the cryogenic rectification process generating at least one liquid stream (154, 156, 160, 162) composed of air or an air-like substance having an argon content no less than air and at least one impure oxygen stream (102, 104) containing oxygen and nitrogen and having an oxygen content no less than that of the air; introducing the at least one impure oxygen stream (102, 104) into a bottom region of an auxiliary column and rectifying the at least one impure oxygen stream (102, 104) within the auxiliary column to form an oxygen containing liquid as a column bottoms and an auxiliary column nitrogen-rich vapor column overhead (140, 142, 144); withdrawing oxygen containing streams (136, 138) from the auxiliary column having a lower nitrogen content of that of the at least one impure oxygen stream (102, 104) and introducing the oxygen containing streams (136, 138) into the lower pressure columns (54, 56) for rectification within the lower pressure columns; and introducing intermediate reflux streams (154, 156, 158) composed of the at least one liquid stream into the lower pressure columns (54, 56) above locations at which the oxygen containing streams (136, 138) are introduced and into the auxiliary column; characterized in that (i) the auxiliary column is operating at substantially the same pressure as the lower pressure column (54, 56); and (ii) the intermediate reflux stream for the auxiliary column is introduced into the auxiliary column above the bottom region thereof. separating air by a cryogenic rectification process employing a plurality of air separation units (3, 4) having higher pressure columns (44, 46) and a lower pressure columns (54, 56) operatively associated with the higher pressure columns (44, 46) to produce oxygen-rich streams (80, 82) that are utilized in producing the oxygen product, the cryogenic rectification process generating at least one liquid stream (154, 156, 160, 162) composed of air or an air-like substance having an argon content no less than air and at least one impure oxygen stream (102, 104) containing oxygen and nitrogen and having an oxygen content no less than that of the air; introducing the at least one impure oxygen stream (102, 104) into a bottom region of an auxiliary column and rectifying the at least one impure oxygen stream (102, 104) within the auxiliary column to form an oxygen containing liquid as a column bottoms and an auxiliary column nitrogen-rich vapor column overhead (140, 142, 144); withdrawing oxygen containing streams (136, 138) from the auxiliary column having a lower nitrogen content of that of the at least one impure oxygen stream (102, 104) and introducing the oxygen containing streams (136, 138) into the lower pressure columns (54, 56) for rectification within the lower pressure columns; and introducing intermediate reflux streams (154, 156, 158) composed of the at least one liquid stream into the lower pressure columns (54, 56) above locations at which the oxygen containing streams (136, 138) are introduced and into the auxiliary column; (i) the auxiliary column is operating at substantially the same pressure as the lower pressure column (54, 56); and (ii) the intermediate reflux stream for the auxiliary column is introduced into the auxiliary column above the bottom region thereof.