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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="seq_1.html#V1">real-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/>

<i><font color="Green">E27</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_1.html#K3" target="_self">^</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#K3" target="_self">^</a> </span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#K3" target="_self">^</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_1.html#K3" target="_self">^</a> </span>)</span>
 ;<br/><a NAME="E2:38_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E3:38_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E4:38_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#K3" target="_self">^</a> </span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E1:38"><i><font color="Green">E27</font></i></a>, <a class="txt" href="rfunct_1.html#E1:38_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_1.html#K3" target="_self">^</a> </span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="txt" href="rfunct_1.html#E1:38_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="txt" href="rfunct_1.html#E3:38_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T15">ABSVALUE:15</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#K3" target="_self">^</a> </span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_1.html#E4:38_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#K3" target="_self">^</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="rfunct_1.html#E1:38"><i><font color="Green">E27</font></i></a>, <a class="txt" href="rfunct_1.html#E1:38_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_1.html#K3" target="_self">^</a>  <a href="hidden.html#R1">=</a>  <a href="seq_1.html#K20">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfunct_1.html#K3" target="_self">^</a> </span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_1.html#E1:38"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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