<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">f</font> be  <a href="seq_1.html#V1">real-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:11_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:11_1_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.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="E3:11_1_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:11_1_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/>
<a NAME="E5:11_1_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a>, <a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/>
<a NAME="E6:11_1_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:11_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:11_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E3:11_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.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="E4:11_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:11_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/><a NAME="E6:11_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/><a NAME="E7:11_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E8:11_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:11"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:11_2"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:11_2_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.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="E3:11_2_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:11_2_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 0 )
 
<b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>;<br/>
<a NAME="E5:11_2_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:11_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:11_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E3:11_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.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="E4:11_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:11_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 0 )
 <b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>;<br/><a NAME="E6:11_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:11_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:11"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K15">-</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
