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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="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2">c=</a>  <a href="mssubfam.html#K2" target="_self">rngs</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <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#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<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="pre_circ.html#D3">PRE_CIRC:def 3</a><br/>

<b>assume </b><a NAME="E3:38"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E5:38"/>
<span class="p1">(<span class="default"><a href="mssubfam.html#K2" target="_self">rngs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mssubfam.html#E3:38"><i><font color="Green">E7</font></i></a>, <a class="ref" href="mssubfam.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E6:38"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mssubfam.html#E1:38"><i><font color="Green">E5</font></i></a>, <a class="txt" href="mssubfam.html#E3:38"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E7:38"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="mssubfam.html#E2:38"><i><font color="Green">E6</font></i></a>, <a class="txt" href="mssubfam.html#E3:38"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:38"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="mssubfam.html#E6:38"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finset_1.html#T27">FINSET_1:27</a></i>;<br/>


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