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<b>let </b><font color="Maroon" title="c1">g</font>, <font color="Maroon" title="c2">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c1">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green" title="E18">A2</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">g</font>
 ;<br/>

<b>thus </b><a NAME="E3:37"/>
<font color="Maroon" title="c2">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="funct_1.html#E1:37"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="funct_1.html#E2:37"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="funct_1.html#T47" target="_self" title="FUNCT_1:th.47">Th47</a></i>;<br/>

<b>assume </b><a NAME="E4:37"/>
not <font color="Maroon" title="c1">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:37"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c3">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">g</font>
 <b>and </b><br/><a NAME="E7:37"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">g</font>
 <b>and </b><br/><a NAME="E8:37"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y2</font>
 <b>and </b><br/><a NAME="E9:37"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c3">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" target="_self" title="FUNCT_1:def.8">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:37"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E12:37"/><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E2:37"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="funct_1.html#E6:37"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" target="_self" title="FUNCT_1:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:37"/><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E15:37"/><i><font color="Green" title="E26">A10</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E2:37"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="funct_1.html#E7:37"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="funct_1.html#D5" target="_self" title="FUNCT_1:def.5">Def5</a></i>;<br/>
<a NAME="E16:37"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="funct_1.html#E2:37"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="funct_1.html#E11:37"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="funct_1.html#E14:37"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="funct_1.html#T23" target="_self" title="FUNCT_1:th.23">Th23</a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<b>hence </b><a NAME="E17:37"/>
contradiction
 <b>by </b><i><a class="txt" href="funct_1.html#E1:37"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="funct_1.html#E8:37"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="funct_1.html#E9:37"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="funct_1.html#E11:37"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="funct_1.html#E12:37"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="funct_1.html#E14:37"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="funct_1.html#E15:37"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="funct_1.html#D8" target="_self" title="FUNCT_1:def.8">Def8</a></i>;<br/>


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