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<b>set </b><font color="Maroon" title="c3">R</font> = <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> ;<br/>
<b>assume </b><a NAME="E1:75_1_1"/>
not <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a>  is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 ;<br/>

<a NAME="E2:75_1_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D9" title="RELAT_1:def.9">RELAT_1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:75_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>
 <b>and </b><br/><a NAME="E5:75_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:75_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="partfun3.html#T2" target="_self" title="PARTFUN3:th.2">Th2</a></i>;<br/>
<a NAME="E7:75_1_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="partfun3.html#E4:75_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> as  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="relat_1.html#D9" title="RELAT_1:def.9">RELAT_1:def 9</a></i>;<br/>
<a NAME="E9:75_1_1"/>
not <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<b>hence </b><a NAME="E10:75_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="partfun3.html#E4:75_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="partfun3.html#E5:75_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>;<br/>


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