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\displaystyle(R(s)|R(x))|e |
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\displaystyle A|e. |
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\epsilon_{\rm el}=\epsilon_{q}(q)-\epsilon_{\lambda}(\lambda). |
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\displaystyle(x|e)|e |
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\displaystyle(x|e)|D(R(x|e)|e)\circ R((x|e)|D(R(x|e)|e))|(R(x|e)|e)\mbox{ by (TC4b${}^{\prime}$)} |
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\displaystyle(x|e)|R(x|e)\circ R((x|e)|R(x|e))|(R(x|e)|e)\mbox{ by (LMU)} |
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\displaystyle(x|e)\circ R(x|e)|(R(x|e)|e)\mbox{ by (TC2)} |
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\displaystyle(x|e)\circ(R(x|e)|R(x|e))|e\mbox{ by (TC6)} |
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\displaystyle(x|e)\circ R(x|e)|e\mbox{ by (2)}, |
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\displaystyle(x|e)\circ R(x|e)|e\mbox{ by (5)} |
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\displaystyle x|D(R(x)|e)\circ A|e\circ R(A|e)|e\mbox{ by (1) and (4)} |
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\displaystyle x|D(R(x)|e)\circ(A|e)|e\mbox{ by (5) with $x$ replaced by $A$} |
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\displaystyle x|D(R(x)|e)\circ A|e\mbox{ by (3)} |
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\displaystyle\rho_{\rm bulk}(q)+\rho^{\rm imp}(q); |
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\displaystyle x|e\mbox{ by (1),} |
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e|x=e|(D(x)\circ x)=e|D(x)\circ R(e|D(x))|x=e\otimes_{l}x. |
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\displaystyle x|e |
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\displaystyle(x|e)|e\mbox{ by Lemma \ref{TC7}} |
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\displaystyle(x|e\circ R(x|e))|e |
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\displaystyle A\circ R(A)|(R(x|e)|e)\mbox{ by (TC4b${}^{\prime}$),} |
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\displaystyle x|e\circ R(x|e)|(R(x|e)|e) |
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\displaystyle x|e\circ(R(x|e)|R(x|e))|e\mbox{ by (TC6)} |
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\displaystyle x|e\circ R(x|e)|e\mbox{, by (2) in the proof of Lemma \ref{TC7}} |
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\displaystyle(x|D(e))\circ R(x|D(e))|e |
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\displaystyle\sigma_{\rm bulk}(\lambda)+\sigma^{\rm imp}(\lambda). |
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\displaystyle x\otimes_{l}e. |
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x\otimes_{l}y=x|D(y)\circ R(x|D(y))|y=x|R(x)\circ R(x|R(x))|y=x\circ R(x)|y=x\circ D(y)|y=x\circ y. |
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e\otimes_{l}e=e|D(e)\circ R(e|D(e))|e=e|R(e)\circ R(e|e)|e=e\circ R(e)|e=R(e)|e=e|e=e, |
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D(x\otimes_{l}y)=D(x|D(y))=D(x|R(x))=D(x), |
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R(x\otimes_{l}y)=R(R(x|D(y))|y)=R(R(x|R(x))|y)=R(R(x)|y)=R(D(y)|y)=R(y). |
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R(xD(y))=R(x|D(y))=D(R(x|D(y))|y)=D(R(xD(y))y), |
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\displaystyle F(s\otimes_{l}t) |
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\displaystyle F(s|D(t)\circ R(s|D(t))|t) |
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\displaystyle F(s|D(t))\circ F(R(s|D(t))|t) |
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\displaystyle F(s)|F(D(t))\circ F(R(s|D(t))|F(t) |
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\displaystyle\rho^{\rm imp}(q) |
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\displaystyle F(s)|D(F(t))\circ R(F(s)|D(F(t)))|F(t) |
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\displaystyle F(s)\otimes_{l}F(t), |
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x|e=(x\circ R(x))|e=x|(R(x)|e)\circ R(x|(R(x)|e))|(R(x)|e), |
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R(x|e)=R(x|(R(x)|e))|(R(x)|e)=R(x|e)|R(R(x)|e)=R(x|e)|(R(x)|e)=(R(x|e)|R(x))|e |
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=(R(x|e)|R(x))|(e|e)=((R(x|e)|R(x))|e)|e=(R(x|e)|(R(x)|e))|e=R(x|e)|e, |
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D(R(x|D(y))|y)=D(R(x|D(y))|D(y))=D(R(x|D(y)))=R(x|D(y)), |
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(s\otimes_{l}t)\otimes_{l}u=((s\otimes_{l}t)|D(u))\circ R((s\otimes_{l}t)|D(u))|u. |
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\displaystyle(s\otimes_{l}t)|D(u) |
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\displaystyle(s|D(t)\circ R(s|D(t))|t)|D(u) |
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\displaystyle(s|D(t))|(D(R(s|D(t))|t|D(u))\circ R((s|D(t))|(D(R(s|D(t))|t|D(u)))|(R(s|D(t))|t|D(u)) |
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\displaystyle\sigma^{\rm imp}(\lambda) |
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\displaystyle A\circ R(A)|(R(s|D(t))|t|D(u)) |
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\displaystyle(s|D(t))|(D(R(s|D(t))|t|D(u))) |
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\displaystyle(s|D(t))|(D(R(s|D(t))|D(t|D(u)))) |
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\displaystyle(s|D(t))|(R(s|D(t))|D(t|D(u))) |
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\displaystyle(s|D(t))|D(t|D(u))\hskip 56.9055pt\mbox{$(*)$} |
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\displaystyle s|(D(t)|D(t|D(u))) |
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\displaystyle s|D(D(t)|D(t|D(u))) |
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\displaystyle s|D(D(t)|(t|D(u)) |
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\displaystyle s|D((D(t)|t)|D(u)) |
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\displaystyle s|D(t|D(u)), |
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\displaystyle-\int^{B}_{-D}dqa_{1}(\lambda-g(q))\rho^{\rm imp}(q), |
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\displaystyle R(A)|(R(s|D(t))|t|D(u)) |
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\displaystyle R(s|D(t|D(u)))|(R(s|D(t))|t|D(u)) |
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\displaystyle R(s|D(t)|D(t|D(u)))|(R(s|D(t))|t|D(u))\mbox{ by $(*)$ above} |
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\displaystyle R((s|D(t)|R(s|D(t)))|D(t|D(u)))|(R(s|D(t))|(D(t|D(u))|(t|D(u))) |
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\displaystyle R((s|D(t)|R(s|D(t)))|D(t|D(u)))|t|D(u) |
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e|f\in D(C) |
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R(x|e)|e=R(x|e) |
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x\in C |
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e,f\in D(C) |
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\displaystyle R((s|D(t))|D(t|D(u)))|t|D(u) |
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\displaystyle p^{\rm imp}_{q}(B) |
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\displaystyle R(A)|t|D(u)\mbox{ by $(*)$ above} |
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\displaystyle R(s|D(t|D(u)))|t|D(u). |
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\displaystyle R((s\otimes_{l}t)|D(u))|u |
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\displaystyle R(R(s|D(t|D(u)))|t|D(u))|u |
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\displaystyle R(R(s|D(t|D(u)))|t|D(u)|R(t|D(u)))|u |
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\displaystyle R(R(s|D(t|D(u)))|t|D(u)|R(t|D(u)))|R(t|D(u))|u |
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\displaystyle R(R(s|D(t|D(u))|t|D(u))|R(t|D(u))|u. |
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\displaystyle(s\otimes_{l}t)\otimes_{l}u |
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\displaystyle((s\otimes_{l}t)|D(u))\circ R((s\otimes_{l}t)|D(u))|u |
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\displaystyle s|D(t|D(u))\circ R(s|D(t|D(u)))|t|D(u)\circ R(R(s|D(t|D(u))|t|D(u))|R(t|D(u))|u. |
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\displaystyle 2\pi\int^{B}_{-D}dq\rho^{\rm imp}(q); |
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\displaystyle s\otimes_{l}(t\otimes_{l}u) |
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\displaystyle(s|D(t\otimes_{l}u))\circ(R(s|D(t\otimes_{l}u))|(t\otimes_{l}u)) |
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\displaystyle s|D(t|D(u))\circ R(s|D(t\otimes_{l}u))|((t|D(u))\circ(R(t|D(u))|u)) |
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\displaystyle s|D(t|D(u))\circ R(s|D(t|D(u)))|t|D(u)\circ R(R(s|D(t|D(u))|t|D(u))|R(t|D(u))|u |
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\displaystyle(s\otimes_{l}t)\otimes_{l}u, |
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s\otimes_{l}t=s\otimes_{r}t=s|D(R(s)|D(t))\circ R(s)|D(t)\circ R(R(s)|D(t))|t. |
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\displaystyle s\otimes_{l}t |
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\displaystyle s|D(t)\circ R(s|D(t))|t |
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\displaystyle(s\circ R(s))|D(t)\circ R(R(s)|D(t))|t |
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\displaystyle s|(D(R(s)|D(t))\circ R(s)|D(t)\circ R(R(s)|D(t))|t, |
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\displaystyle p^{\rm imp}_{\lambda}(Q) |
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s\otimes t=s|(D(R(s)|D(t))\circ R(s)|D(t)\circ R(R(s)|D(t))|t. |
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\displaystyle(a\otimes b)\otimes c |
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\displaystyle(a\otimes_{r}b)\otimes_{l}c\mbox{ by Proposition \ref{lrequal}} |
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\displaystyle(a\otimes_{r}b)|D(c)\circ R((a\otimes_{r}b)|D(c))|c |
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\displaystyle(a|D(R(a)|b)\circ R(a)|b)|D(c)\circ R((a\otimes b)|D(c))|c\mbox{ by (TC7b)} |
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\displaystyle a|D(R(a)|b|D(c))\circ R(a)|b|D(c)\circ R(R(a)|b|D(c))|c, |
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s\otimes t=s|D(t)\circ R(s)|t\mbox{ for all }s,t\in C. |
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\displaystyle x|D(y)\circ R(x)|y |
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\displaystyle(x\circ R(x))|D(y)\circ R(x)|(D(y)\circ y) |