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<div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:3"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="binop_2.html#K33">addreal</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:3"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E3:3"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <a href="binop_2.html#K33">addreal</a> 
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<i><font color="Green">E41</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="finseqop.html#K1">.:</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#D4">RVSUM_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T84">FUNCOP_1:84</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
;<br/>
<i><font color="Green">E42</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
;<br/>
<a NAME="E6:3"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:3_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span>
 ;<br/>

<i><font color="Green">E44</font></i>: <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
0
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E3:3_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, , , <a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a></i>;<br/>
<b>hence </b><a NAME="E4:3_3"/>
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:3"/>
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div>
