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

<b>let </b><font color="Maroon" title="c1">f</font> be  <a href="seq_1.html#NV1" title="SEQ_1:NV.1">real-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>

<i><font color="Green" title="E21">A1</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">^</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#D8" target="_self" title="RFUNCT_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self" title="RFUNCT_1:th.15">Th15</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">^</a> </span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#D8" target="_self" title="RFUNCT_1:def.8">Def8</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">^</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:33_2"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">^</a> </span>)</span>
 ;<br/><a NAME="E2:33_2"/><b>then </b><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">^</a> </span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E1:33"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">^</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
<b>by </b><i><a class="txt" href="rfunct_1.html#E1:33_2"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="rfunct_1.html#D8" target="_self" title="RFUNCT_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
<b>by </b><i><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></i>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217" title="XCMPLX_1:th.217">XCMPLX_1:217</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T15" title="ABSVALUE:th.15">ABSVALUE:15</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217" title="XCMPLX_1:th.217">XCMPLX_1:217</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">^</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">c</font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_1.html#E2:33_2"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="rfunct_1.html#D8" target="_self" title="RFUNCT_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">^</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">c</font>
<b>by </b><i><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:33"/>
<span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">^</a> </span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E1:33"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
