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<b>let </b><font color="Maroon">S</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>



<b>assume </b><a NAME="E1:13"/>
<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="pencil_2.html#R1" target="_self">are_joinable</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:13"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:13"/><i><font color="Green">E33</font></i>: 
( ( for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>  st <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> holds <br/>( <font color="Olive">W</font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Olive">W</font> is <a href="pencil_1.html#V2">strong</a> ) ) &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/>2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E5:13"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="hidden.html#M1">set</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">d1</font>,<font color="Olive">d2</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g</font> being  <a href="funct_1.html#V2">one-to-one</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E7:13"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> )
 <b>and </b><br/><a NAME="E8:13"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E9:13"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="pencil_2.html#S1" target="_self">PENCIL_2:sch 1</a>(, );<br/></i>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E10:13"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> &amp; ( for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>  st <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> holds <br/>( <font color="Olive">W</font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Olive">W</font> is <a href="pencil_1.html#V2">strong</a> ) ) &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> holds <br/>2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i>, , , </i>;<br/>


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