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<b>let </b><font color="Maroon">A</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">B</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a>  <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">A</font>;<br/>



<a NAME="E1:62"/><i><font color="Green">E25</font></i>: 
( 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> &amp; the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="altcat_2.html#R2">cc=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> &amp; the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="altcat_2.html#R2">cc=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a></i>;<br/>
<a NAME="E2:62"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">A</font>,<font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a>  <a href="yellow18.html#R2">are_opposite</a>  &amp; <font color="Maroon">B</font>,<font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a>  <a href="yellow18.html#R2">are_opposite</a>  )
 
<b>by </b><i><a class="ref" href="yellow18.html#D4">YELLOW18:def 4</a></i>;<br/>
<a NAME="E3:62"/><b>then </b><i><font color="Green">E33</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </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="yellow18.html#D3">YELLOW18:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:62"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span>
 <b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a><br/>

<a NAME="E5:62"/>
( the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="altcat_2.html#K1">~</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> &amp; the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="altcat_2.html#K1">~</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow18.html#D3">YELLOW18:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:62"/>
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="altcat_2.html#R2">cc=</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span>
 <b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a>, <a class="txt" href="yellow20.html#E3:62"><i><font color="Green">E33</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:62"/>
<span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a>, , <a class="ref" href="mcart_1.html#T77">MCART_1:77</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D2">ALTCAT_2:def 2</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E8:62"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:62"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x1</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">x2</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">x3</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">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="mcart_1.html#K3">]</a></span> )
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T72">MCART_1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x1</font>, <font color="Maroon">b</font> = <font color="Maroon">x2</font>, <font color="Maroon">c</font> = <font color="Maroon">x3</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="yellow20.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font>, <font color="Maroon">c1</font> = <font color="Maroon">c</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a>, <a class="ref" href="yellow20.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow18.html#D3">YELLOW18:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1'</font> = <font color="Maroon">a1</font>, <font color="Maroon">b1'</font> = <font color="Maroon">b1</font>, <font color="Maroon">c1'</font> = <font color="Maroon">c1</font> as   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow18.html#D3">YELLOW18:def 3</a></i>;<br/>
<a NAME="E15:62"/><i><font color="Green">E39</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K1">~</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow18.html#D3">YELLOW18:def 3</a></i>;<br/>
<a NAME="E16:62"/><i><font color="Green">E51</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">c1'</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K1">~</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow18.html#D3">YELLOW18:def 3</a></i>;<br/>
<a NAME="E17:62"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">x3</font>,<font color="Maroon">x2</font>,<font color="Maroon">x1</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow20.html#T5" target="_self">Th5</a>, <a class="ref" href="mcart_1.html#T73">MCART_1:73</a></i>;<br/>
<a NAME="E18:62"/><i><font color="Green">E53</font></i>: 
( the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a1'</font>,<font color="Maroon">b1'</font>,<font color="Maroon">c1'</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="yellow20.html#T5" target="_self">Th5</a>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E19:62"/>
( the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</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">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="mcart_1.html#K3">]</a></span> &amp; the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> 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">c1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a1</font></span><a href="mcart_1.html#K3">]</a></span> )
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E20:62"/><b>then </b>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">B</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c1</font>,<font color="Maroon">b1</font>,<font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a>, , <a class="ref" href="altcat_2.html#D2">ALTCAT_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E21:62"/>
the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U2">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="yellow20.html#E15:62"><i><font color="Green">E39</font></i></a>, <a class="ref" href="yellow20.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow20.html#T10" target="_self">Th10</a>, <a class="txt" href="yellow20.html#E2:62"><i><font color="Green">E32</font></i></a></i>;<br/>


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