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<div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1"/>
(  not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1_1"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_1_1_1_1"/>
 <a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="ref" href="finseq_6.html#D2">FINSEQ_6:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a><br/><b>assume </b><b>that </b><br/><a NAME="E2:30_1_1_1_2"/><i><font color="Green">E23</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:30_1_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E4:30_1_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:30_1_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:30_1_1_1_2"/><i><font color="Green">E27</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="revrot_1.html#T14" target="_self">Th14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2_1"><b>now </b></a><div class="add"><a NAME="E1:30_1_1_1_2_1"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:30_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_5.html#T53">FINSEQ_5:53</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1_2_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:30_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:30_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="revrot_1.html#E2:30_1_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="revrot_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:30_1_1_1_2_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:30_1_1_1_2_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="revrot_1.html#E2:30_1_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1

;<br/><a NAME="E5:30_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:30_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="revrot_1.html#E4:30_1_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="revrot_1.html#E3:30_1_1_1_2_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="revrot_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E6:30_1_1_1_2_1_1_1"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E7:30_1_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:30_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/><a NAME="E8:30_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E6:30_1_1_1_2_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:30_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:30_1_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="revrot_1.html#E1:30_1_1_1_2_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E10:30_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E11:30_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:30_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="revrot_1.html#E2:30_1_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><b>hence </b><a NAME="E13:30_1_1_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="revrot_1.html#E2:30_1_1_1_2_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="revrot_1.html#E5:30_1_1_1_2_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="revrot_1.html#E8:30_1_1_1_2_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_1_2_1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:30_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="revrot_1.html#E3:30_1_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="revrot_1.html#E5:30_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="revrot_1.html#E6:30_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:30_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:30_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="revrot_1.html#E1:30_1_1_1_2_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="revrot_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E4:30_1_1_1_2_1_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E5:30_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="revrot_1.html#E1:30_1_1_1_2_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:30_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:30_1_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:30_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="revrot_1.html#E6:30_1_1_1_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="binarith.html#T56">BINARITH:56</a></i>
;<br/><a NAME="E8:30_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:30_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="revrot_1.html#E3:30_1_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="revrot_1.html#E6:30_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="revrot_1.html#E4:30_1_1_1_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="revrot_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E9:30_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:30_1_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="revrot_1.html#E1:30_1_1_1_2_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E10:30_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><a NAME="E11:30_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E12:30_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:30_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/><a NAME="E13:30_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E3:30_1_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="revrot_1.html#E6:30_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E14:30_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E15:30_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E16:30_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="revrot_1.html#E6:30_1_1_1_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>hence </b><a NAME="E17:30_1_1_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="revrot_1.html#E3:30_1_1_1_2_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="revrot_1.html#E8:30_1_1_1_2_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="revrot_1.html#E11:30_1_1_1_2_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:30_1_1_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
