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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b>
 <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">F<sub>1</sub></font>()
;<br/><b>thus </b><a NAME="E2:1_1_1_1"/>
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">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="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font>] ) )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_2_1"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_2_1_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:1_1_1_2_1_1_1_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:1_1_1_2_1_1_1_2"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_1_2_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:1_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> 1 <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E4:1_1_1_2_1_1_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">M</font> =  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E4:1_1_1_2"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font>;<br/>

<a NAME="E1:1_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>()
 
;<br/>
<b>hence </b><a NAME="E2:1_1_1_2_2"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E6:1_1_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><a NAME="E7:1_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">f</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_2_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:1_1_1_2_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_2_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E10:1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>take </b>
<font color="Maroon">q</font>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_2_4"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/><a NAME="E2:1_1_1_2_4"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:1_1_1_2_4"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_2_4"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:1_1_1_2"/>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">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="finseq_1.html#K2">Seg</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font>] ) )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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