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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>thus </b><i><font color="Green">E43</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> = 
 <a href="funct_6.html#K4" target="_self">meet</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#T49" target="_self">Th49</a></i>
<br/>.= 
 <a href="funct_6.html#K4" target="_self">meet</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i><a class="ref" href="funct_6.html#T34" target="_self">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#T40" target="_self">Th40</a></i>
;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:68"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:68"/><b>then </b>
( <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" target="_self">rngs</a> <span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="funct_6.html#E1:68"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_6.html#T49" target="_self">Th49</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:68"/><b>then </b>
( <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" target="_self">rngs</a> <span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span>)</span> &amp;  <a href="funct_6.html#K3" target="_self">rngs</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="funct_6.html#T34" target="_self">Th34</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:68"/><i><font color="Green">E45</font></i>: 
( <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E7:68"/><i><font color="Green">E46</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="funct_6.html#E2:68"><i><font color="Green">E44</font></i></a>, <a class="txt" href="funct_6.html#E6:68"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, <a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E9:68"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span>)</span> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span>)</span> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span>)</span> <a href="binop_1.html#K1">.</a> 2,<font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span>)</span> <a href="binop_1.html#K1">.</a> 2,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="txt" href="funct_6.html#E1:68"><i><font color="Green">E43</font></i></a>, <a class="txt" href="funct_6.html#E2:68"><i><font color="Green">E44</font></i></a>, <a class="txt" href="funct_6.html#E6:68"><i><font color="Green">E45</font></i></a>, <a class="txt" href="funct_6.html#E7:68"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_6.html#T51" target="_self">Th51</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="funct_5.html#T45">FUNCT_5:45</a></i>;<br/>
<b>hence </b><a NAME="E10:68"/>
<span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="funct_6.html#E6:68"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>


</div>
