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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green">E3</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<a NAME="E2:4"/><i><font color="Green">E4</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> is <a href="relat_1.html#V1">Relation-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a><br/>

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:4_1"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E3:4_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:4_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E3:4_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:4"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V1">Function-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a><br/>

<b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E5</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_2"/><i><font color="Green">E6</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:4_2"/><i><font color="Green">E7</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:4_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="wellfnd1.html#E3:4_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="wellfnd1.html#E5:4_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/>
<a NAME="E7:4_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E1:4"><i><font color="Green">E3</font></i></a>, <a class="txt" href="wellfnd1.html#E3:4_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="wellfnd1.html#E5:4_2"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:4_2"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T131">PARTFUN1:131</a></i>;<br/>
<b>thus </b><a NAME="E10:4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:4_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="wellfnd1.html#E3:4_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="wellfnd1.html#E5:4_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="wellfnd1.html#E9:4_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:4"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E2:4"><i><font color="Green">E4</font></i></a></i>;<br/>


</div>
