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<b>let </b><font color="Maroon" title="c1">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">g</font>, <font color="Maroon" title="c3">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_1"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c3">f</font>
 ;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_1_1_1"/>
<font color="Maroon" title="c4">x</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">g</font>
 ;<br/><a NAME="E2:70_1_1_1"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D4" target="_self" title="FUNCT_1:def.4">Def4</a></i>;<br/><a NAME="E3:70_1_1_1"/><b>then </b><i><font color="Green" title="E34">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E1:70_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="relat_1.html#D12" title="RELAT_1:def.12">RELAT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E4:70_1_1_1"/>
<font color="Maroon" title="c4">x</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="c3">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/><a NAME="E5:70_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E3:70_1_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_1.html#D4" target="_self" title="FUNCT_1:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E6:70_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E1:70_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="funct_1.html#E2:70_1_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="relat_1.html#D12" title="RELAT_1:def.12">RELAT_1:def 12</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:70_1_1"/>
<font color="Maroon" title="c4">x</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="c3">f</font>
 ;<br/><a NAME="E3:70_1_1"/><b>then </b><i><font color="Green" title="E33">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D4" target="_self" title="FUNCT_1:def.4">Def4</a></i>;<br/><b>assume </b><a NAME="E4:70_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 ;<br/><a NAME="E5:70_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E1:70_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="funct_1.html#E3:70_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="relat_1.html#D12" title="RELAT_1:def.12">RELAT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E6:70_1_1"/>
<font color="Maroon" title="c4">x</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">g</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/></div>
<b>end;</b></div><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E3:70_1"/>
<font color="Maroon" title="c4">x</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">g</font>
 ;<br/><a NAME="E4:70_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D4" target="_self" title="FUNCT_1:def.4">Def4</a></i>;<br/><a NAME="E5:70_1"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E1:70_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="relat_1.html#D12" title="RELAT_1:def.12">RELAT_1:def 12</a></i>;<br/><a NAME="E6:70_1"/><b>then </b>
<font color="Maroon" title="c4">x</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="c3">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E7:70_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E5:70_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="funct_1.html#D4" target="_self" title="FUNCT_1:def.4">Def4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:70"/><i><font color="Green" title="E32">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</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">g</font> iff ( <font color="Olive" title="b1">x</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="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> ) )
 <b>and </b><br/><a NAME="E3:70"/><i><font color="Green" title="E33">A7</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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">g</font> holds <br/><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_2_1"/><i><font color="Green" title="E34">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">g</font>
 ;<br/><a NAME="E2:70_2_1"/><b>then </b><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c4">x</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">g</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/><b>then </b><i><font color="Green" title="E36">A10</font></i>: <font color="Maroon" title="c5">y</font> = 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="funct_1.html#E1:70_2_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="funct_1.html#D4" target="_self" title="FUNCT_1:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="funct_1.html#E3:70"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="funct_1.html#E2:70_2_1"><i><font color="Green" title="E35">A9</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:70_2_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E2:70"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="funct_1.html#E2:70_2_1"><i><font color="Green" title="E35">A9</font></i></a></i>;<br/><a NAME="E5:70_2_1"/>
<font color="Maroon" title="c4">x</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="c3">f</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E2:70"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="funct_1.html#E2:70_2_1"><i><font color="Green" title="E35">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E6:70_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E3:70_2_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="funct_1.html#D4" target="_self" title="FUNCT_1:def.4">Def4</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:70_2"/><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 ;<br/><b>assume </b><a NAME="E3:70_2"/><i><font color="Green" title="E35">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font>
 ;<br/><a NAME="E4:70_2"/><b>then </b><i><font color="Green" title="E36">A13</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8" target="_self" title="FUNCT_1:th.8">Th8</a></i>;<br/><a NAME="E5:70_2"/>
<font color="Maroon" title="c4">x</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="c3">f</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E3:70_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/><a NAME="E6:70_2"/><b>then </b><i><font color="Green" title="E37">A14</font></i>: 
<font color="Maroon" title="c4">x</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">g</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E2:70"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="funct_1.html#E2:70_2"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="funct_1.html#E4:70_2"><i><font color="Green" title="E36">A13</font></i></a></i>;<br/><a NAME="E7:70_2"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E3:70"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="funct_1.html#E4:70_2"><i><font color="Green" title="E36">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E8:70_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="funct_1.html#E6:70_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="funct_1.html#D4" target="_self" title="FUNCT_1:def.4">Def4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:70"/>
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D12" title="RELAT_1:def.12">RELAT_1:def 12</a></i>;<br/>


</div>
