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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  implies  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b3">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b4">b9</font>,<font color="Olive" title="b3">a9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">A</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow18.html#D3" target="_self" title="YELLOW18:def.3">YELLOW18:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> ex <font color="Olive" title="b4">a9</font>, <font color="Olive" title="b5">b9</font>, <font color="Olive" title="b6">c9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font> st <br/>( <font color="Olive" title="b4">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b5">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b6">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  not  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">B</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Olive" title="b4">a9</font>,<font color="Olive" title="b5">b9</font>,<font color="Olive" title="b6">c9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K1" title="FUNCTOR0:func.1">~</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>)</span>)</span> ) or  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b3">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b4">b9</font>,<font color="Olive" title="b3">a9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:26"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b4">a9</font>, <font color="Olive" title="b5">b9</font>, <font color="Olive" title="b6">c9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b4">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b5">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b6">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> holds <br/> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">B</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Olive" title="b4">a9</font>,<font color="Olive" title="b5">b9</font>,<font color="Olive" title="b6">c9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K1" title="FUNCTOR0:func.1">~</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b3">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b4">b9</font>,<font color="Olive" title="b3">a9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">b9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Olive" title="b2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> holds <br/><span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b9</font>,<font color="Olive" title="b1">a9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a9</font>, <font color="Maroon" title="c6">b9</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c6">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c5">a9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )</span><br/>



<span class="kw">thus </span><a NAME="E4:26"/>
( <font color="Maroon" title="c5">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c6">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c5">a9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="altcat_2.html#T6" title="ALTCAT_2:th.6">ALTCAT_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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