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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>assume </b><b>that </b><br/><a NAME="E1:55"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K7" target="_self">sproduct</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:55"/><i><font color="Green">E32</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="E3:55"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</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> holds <br/><font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 
<b>by </b><i><a class="txt" href="ami_1.html#E1:55"><i><font color="Green">E31</font></i></a>, <a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="ami_1.html#E2:55"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ami_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E5:55"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1"/>
<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>3</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="E3:55_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1: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="E5:55_1"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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>
 <b>and </b><br/><a NAME="E6:55_1"/><i><font color="Green">E36</font></i>: 
<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="ami_1.html#E3:55_1"><i><font color="Green">E34</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>6</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="ami_1.html#E1:55"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ami_1.html#E6:55_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/>
<a NAME="E8:55_1"/><i><font color="Green">E37</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>7</sub></font>
 
<b>by </b><i><a class="txt" href="ami_1.html#E5:55_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E9:55_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</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="ami_1.html#E1:55"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ami_1.html#E6:55_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ami_1.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E10:55_1"/>
<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="ami_1.html#E8:55_1"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:55_2"/>
<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>3</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="E3:55_2"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1: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="E5:55_2"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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>
 <b>and </b><br/><a NAME="E6:55_2"/><i><font color="Green">E36</font></i>: 
<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="ami_1.html#E3:55_2"><i><font color="Green">E34</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>6</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="ami_1.html#E1:55"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ami_1.html#E6:55_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/><a NAME="E8:55_2"/><i><font color="Green">E37</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>7</sub></font>
 <b>by </b><i><a class="txt" href="ami_1.html#E5:55_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="ami_1.html#E5:55_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</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="ami_1.html#E3:55_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
;<br/><b>hence </b><a NAME="E10:55_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</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="ami_1.html#E1:55"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ami_1.html#E6:55_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ami_1.html#E8:55_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="ami_1.html#T25" target="_self">Th25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:55"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K7" target="_self">sproduct</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ami_1.html#E5:55"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ami_1.html#D16" target="_self">Def16</a></i>;<br/>


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