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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>thus </b><a NAME="E1:5"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:5_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:5_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:5_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:5_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="functor0.html#E4:5_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E8:5_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="functor0.html#E4:5_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="functor0.html#E7:5_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E9:5_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="functor0.html#E3:5_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="functor0.html#E7:5_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_4.html#T43">FUNCT_4:43</a></i>;<br/>
<a NAME="E10:5_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="functor0.html#E5:5_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="functor0.html#E7:5_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="functor0.html#E8:5_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_4.html#T44">FUNCT_4:44</a></i>;<br/>
<b>hence </b><a NAME="E11:5_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="functor0.html#E4:5_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="functor0.html#E9:5_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:5"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:5"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:5"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:5"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="functor0.html#E5:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E9:5"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="functor0.html#E5:5"><i><font color="Green">E5</font></i></a>, <a class="txt" href="functor0.html#E8:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E10:5"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="functor0.html#E4:5"><i><font color="Green">E4</font></i></a>, <a class="txt" href="functor0.html#E8:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_4.html#T43">FUNCT_4:43</a></i>;<br/>
<a NAME="E11:5"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="functor0.html#E6:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="functor0.html#E8:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="functor0.html#E9:5"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>;<br/>
<b>hence </b><a NAME="E12:5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="functor0.html#E5:5"><i><font color="Green">E5</font></i></a>, <a class="txt" href="functor0.html#E10:5"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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