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<b>let </b><font color="Maroon" title="c1">f</font> be  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/> for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">g</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">i2</font> holds <br/> <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b1">g</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i1</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i2</font></span><br/><b>let </b><font color="Maroon" title="c2">g</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c2">g</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Olive" title="b1">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">i2</font> holds <br/> <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">g</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i1</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i2</font></span><br/><b>let </b><font color="Maroon" title="c3">i1</font>, <font color="Maroon" title="c4">i2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">g</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i2</font> implies  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">g</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> )</span><br/>





<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E47">A1</font></i>: 
( <font color="Maroon" title="c2">g</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">g</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span><br/>

<a NAME="E2:48"/><b>then </b><i><font color="Green" title="E48">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c3">i1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font> )
 
<b>by </b><i><a class="ref" href="jordan4.html#T37" target="_self" title="JORDAN4:th.37">Th37</a></i>;<br/>
<a NAME="E3:48"/><i><font color="Green" title="E49">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i1</font> &amp; <font color="Maroon" title="c3">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c4">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> holds <br/><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="jordan4.html#K1" title="JORDAN4:func.1">S_Drop</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c1">f</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="jordan4.html#E1:48"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="jordan4.html#D2" target="_self" title="JORDAN4:def.2">Def2</a></i>;<br/>
<a NAME="E4:48"/><b>then </b><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="topreal1.html#NR2" title="TOPREAL1:NR.2">is_S-Seq</a> 
 
<b>by </b><i><a class="txt" href="jordan4.html#E1:48"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="jordan4.html#E2:48"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="jordan4.html#T51" target="_self" title="JORDAN4:th.51">Th51</a></i>;<br/>
<a NAME="E5:48"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="jordan4.html#E3:48"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:48"/><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="jordan4.html#E3:48"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:48"/><b>then </b><i><font color="Green" title="E53">A7</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i1</font>
 
<b>by </b><i><a class="txt" href="jordan4.html#E2:48"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="jordan4.html#E3:48"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="jordan4.html#E5:48"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E8:48"/><i><font color="Green" title="E54">A8</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i1</font>
 
<b>by </b><i><a class="txt" href="jordan4.html#E3:48"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="jordan4.html#E5:48"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E9:48"/>
<font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan4.html#E3:48"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:48"/><b>then </b>
( <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan4.html#E3:48"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="jordan4.html#E6:48"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="jordan4.html#E7:48"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="jordan4.html#E8:48"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<b>hence </b><a NAME="E11:48"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">g</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:48"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="topreal4.html#D1" title="TOPREAL4:def.1">TOPREAL4:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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