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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><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/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<i><font color="Green">E29</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">r</font> <a href="seq_1.html#K12">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></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">r</font> <a href="seq_1.html#K12">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
<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">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K12">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></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="57_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>assume </b><a NAME="E1:57_2"/><i><font color="Green">E30</font></i>: 
<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"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K12">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span>
 ;<br/><a NAME="E2:57_2"/><b>then </b>
<font color="Maroon">c</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">r</font> <a href="seq_1.html#K12">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E3:57_2"/><b>then </b><i><font color="Green">E31</font></i>: 
( <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"><font color="Maroon">r</font> <a href="seq_1.html#K12">(#)</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:57_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>
 <b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/><a NAME="E5:57_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T11" target="_self">Th11</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:57_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<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"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E7:57_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<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"><font color="Maroon">r</font> <a href="seq_1.html#K12">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i><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">r</font> <a href="seq_1.html#K12">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K12">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</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>
<b>by </b><i><a class="ref" href="rfunct_1.html#T11" target="_self">Th11</a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K12">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T13" target="_self">Th13</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="E3:57"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K12">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="seq_1.html#K12">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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