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<b>let </b><font color="Maroon">B</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A</font>;<br/>

<b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM8">ManySortedSet</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:56_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">a1</font>, <font color="Olive">a2</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> holds  <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">a2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a2</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow18.html#D7">YELLOW18:def 7</a></i>;<br/>
<b>set </b><font color="Maroon">G</font> = <font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>;<br/>
<a NAME="E3:56_1_1"/><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:56_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> as   <a href="struct_0.html#NM8">ManySortedSet</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">G</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D7">YELLOW18:def 7</a><br/>

<b>let </b><font color="Maroon">a1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>, <font color="Maroon">a2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>;<br/>

<a NAME="E6:56_1_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">a1</font>, <font color="Maroon">b2</font> = <font color="Maroon">a2</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E8:56_1_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E9:56_1_1"/><b>then </b>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/>
<b>hence </b><a NAME="E10:56_1_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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