[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that the orthogonal easy free quantum groups are

[[math]] \xymatrix@R=1pt@C=100pt{ H_N^+\ar[r]&O_N^+\\ \ &\\ \ &\\ &B_N''^+\ar[uuu]\\ S_N'^+\ar[uuuu]&\\ &B_N'^+\ar[uu]\\ \ &\\ \ &\\ S_N^+\ar[r]\ar[uuuu]&B_N^+\ar[uuu]} [[/math]]

where [math]S_N'^+=S_N^+\times\mathbb Z_2[/math], and where [math]B_N'^+\subset B_N''^+[/math] are easy liberations of [math]B_N'=B_N\times\mathbb Z_2[/math].