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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green">E15</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
 ;<br/>

<a NAME="E2:24"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E3:24"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:24"><i><font color="Green">E15</font></i></a>, <a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E4:24"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E2:24"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:24_1"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:24_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:24_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_2.html#E1:24"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E4:24_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a></i>;<br/><a NAME="E5:24_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:24_1"/><b>then </b><i><font color="Green">E19</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E7:24_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rfunct_2.html#E4:24"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rfunct_2.html#E1:24_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rfunct_2.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E8:24_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rfunct_2.html#E6:24_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:24"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>


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