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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>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="E2:27"/><i><font color="Green">E6</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>2</sub></font> holds <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#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S4">FUNCT_1:sch 4</a>();<br/></i>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E4:27"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:27_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:27_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:27_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pre_circ.html#E2:27"><i><font color="Green">E6</font></i></a>, <a class="txt" href="pre_circ.html#E1:27_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pre_circ.html#E2:27_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="pre_circ.html#E3:27"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pre_circ.html#E3:27_1"><i><font color="Green">E10</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="pre_circ.html#E3:27"><i><font color="Green">E7</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:27_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:27"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pre_circ.html#E2:27"><i><font color="Green">E6</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:27"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:27_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pre_circ.html#E2:27"><i><font color="Green">E6</font></i></a>, <a class="txt" href="pre_circ.html#E3:27_2"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<a NAME="E7:27_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E3:27"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E8:27_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/>
<b>hence </b><a NAME="E9:27_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="pre_circ.html#E4:27_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pre_circ.html#E7:27_2"><i><font color="Green">E10</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E7:27"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:27"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E11:27"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="pre_circ.html#E9:27"><i><font color="Green">E8</font></i></a>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/>
<a NAME="E12:27"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="pre_circ.html#E9:27"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pre_circ.html#E11:27"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pre_circ.html#E9:27"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pre_circ.html#E11:27"><i><font color="Green">E9</font></i></a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<a NAME="E14:27"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E3:27"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pre_circ.html#E11:27"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:27"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pre_circ.html#E2:27"><i><font color="Green">E6</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div>
