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<b>let </b><font color="Maroon">A</font> be    <a href="altcat_1.html#L2">AltCatStr</a> , <font color="Maroon">B</font> be    <a href="altcat_1.html#L2">AltCatStr</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="yellow20.html#R1" target="_self">have_the_same_composition</a> 
 ;<br/><b>let </b><font color="Maroon">a1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">a2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">a3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E2:11_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E3:11_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span>
 ;<br/><a NAME="E4:11_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span></span>)</span> &amp; the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span> <a href="partfun1.html#R1">tolerates</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span> )
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:11_1"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font>, <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">a2</font>,<font color="Olive">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">a2</font>,<font color="Olive">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">a2</font>,<font color="Olive">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">a2</font>,<font color="Olive">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 ;<br/>

<b>let </b><font color="Maroon">a1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">a2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">a3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a>,<a class="ref" href="yellow20.html#D1" target="_self">YELLOW20:def 1</a><br/>

<b>assume </b><a NAME="E3:11"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:11"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:11"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font></span><a href="mcart_1.html#K3">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


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