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<b>let </b><font color="Maroon">p</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E53</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:27"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">q</font>
;<br/>

<a NAME="E5:27"/>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E6:27"/><b>then </b>
1 <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E7:27"/><b>then </b><i><font color="Green">E57</font></i>: 
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; 0 <a href="hidden.html#R2">in</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E8:27"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0 )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E9:27"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font>
 <b>by </b><i>, , </i>;<br/>

<a NAME="E10:27"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , , <a class="ref" href="afinsq_1.html#T28">AFINSQ_1:28</a></i>;<br/>
<a NAME="E11:27"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>assume </b><a NAME="E12:27"/>
not <font color="Maroon">q</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:27"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E15:27"/>
<font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E16:27"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0 )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E17:27"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:27"/><b>then </b>
( 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i>, , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E19:27"/><b>then </b>
1 <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>
<a NAME="E20:27"/><b>then </b>
1 <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E21:27"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
