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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/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<i><font color="Green">E26</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K12">(#)</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"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K12">(#)</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"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K12">(#)</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K12">(#)</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"><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="txt" href="rfunct_1.html#E1:36"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rfunct_1.html#T17" target="_self">Th17</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="seq_1.html#D6">SEQ_1:def 6</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="seq_1.html#K12">(#)</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#D6">SEQ_1:def 6</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</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"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K12">(#)</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="E2:36_2"/><b>then </b>
<font color="Maroon">c<sub>3</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"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K12">(#)</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"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K12">(#)</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="E3:36_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</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>2</sub></font> <a href="seq_1.html#K12">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E4:36_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</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#E2:36"><i><font color="Green">E26</font></i></a>, <a class="txt" href="rfunct_1.html#E1:36_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K12">(#)</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>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K12">(#)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="txt" href="rfunct_1.html#E1:36_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</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>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> 
<b>by </b><i><a class="txt" href="rfunct_1.html#E3:36_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xcmplx_0.html#K3">*</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>3</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T205">XCMPLX_1:205</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xcmplx_0.html#K3">*</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>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_1.html#E4:36_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="seq_1.html#K12">(#)</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>3</sub></font>
<b>by </b><i><a class="txt" href="rfunct_1.html#E2:36"><i><font color="Green">E26</font></i></a>, <a class="txt" href="rfunct_1.html#E1:36_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:36"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K12">(#)</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="seq_1.html#K12">(#)</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#E2:36"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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