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



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="int_1.html#NM1">Integer</a>] <b>means </b> ex <font color="Olive">i0</font>, <font color="Olive">j0</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">i0</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">j0</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 );<br/>
<b>assume </b><a NAME="E1:15"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<a NAME="E2:15"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/>
( not <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; not <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:15_1"/><b>then </b>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 
;<br/>
<a NAME="E3:15_1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
;<br/>
<b>hence </b><a NAME="E4:15_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:15"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/>
<a NAME="E4:15"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E5:15"/><b>then </b><i><font color="Green">E31</font></i>: 
 ex <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="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:15"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E8:15"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>thus </b><a NAME="E9:15"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  ex <font color="Olive">i0</font>, <font color="Olive">j0</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">i0</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">j0</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) &amp; ( for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st  ex <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> ) )
 <b>by </b><i>, </i>;<br/>


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