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<b>consider </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> </span>)</span> <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_3.html#K5">--&gt;</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E3:36_1_1"/><i><font color="Green">E20</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_3.html#K5">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_3.html#K5">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_3.html#K5">--&gt;</a> <font color="Maroon">c<sub>2</sub></font> as  <a href="comput_1.html#V2" target="_self">from-natural-fseqs</a> <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="comput_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<a NAME="E5:36_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="comput_1.html#V4" target="_self">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D4" target="_self">COMPUT_1:def 4</a><br/>

<b>assume </b><a NAME="E1:36_1_1_1"/>
( <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>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:36_1_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:36_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<b>hence </b><a NAME="E3:36_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:36_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T55">SUBSET_1:55</a></i>;<br/>
<a NAME="E7:36_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:36_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:36_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="comput_1.html#V3" target="_self">len-total</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D3" target="_self">COMPUT_1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:36_1_1_2"/><i><font color="Green">E23</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E2:36_1_1_2"/><i><font color="Green">E24</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>5</sub></font>
 ;<br/>

<a NAME="E3:36_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E4:36_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:36_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="comput_1.html#E2:36_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:36_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="comput_1.html#E1:36_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="comput_1.html#E3:36_1_1_2"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:36_1_1"/>
( not <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="seqm_3.html#V6">to-naturals</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="comput_1.html#V3" target="_self">len-total</a> )
 <b>by </b><i><a class="txt" href="comput_1.html#E5:36_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="comput_1.html#E7:36_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/>


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