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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>  st <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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <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/> for <font color="Olive" title="b5">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b6">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b3">a9</font>  st <font color="Olive" title="b6">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> holds <br/>( <font color="Olive" title="b5">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  iff <font color="Olive" title="b6">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:45"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>  st <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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <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/> for <font color="Olive" title="b5">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b6">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b3">a9</font>  st <font color="Olive" title="b6">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> holds <br/>( <font color="Olive" title="b5">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  iff <font color="Olive" title="b6">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  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/> for <font color="Olive" title="b3">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font><br/>  for <font color="Olive" title="b4">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b9</font>,<font color="Olive" title="b1">a9</font>  st <font color="Olive" title="b4">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> holds <br/>( <font color="Olive" title="b3">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  iff <font color="Olive" title="b4">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E3:45"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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/> for <font color="Olive" title="b3">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font><br/>  for <font color="Olive" title="b4">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b9</font>,<font color="Olive" title="b1">a9</font>  st <font color="Olive" title="b4">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> holds <br/>( <font color="Olive" title="b3">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  iff <font color="Olive" title="b4">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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  for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font><br/>  for <font color="Olive" title="b2">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c5">a9</font>  st <font color="Olive" title="b2">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  iff <font color="Olive" title="b2">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:45"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c5">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E5:45"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c6">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font><br/>  for <font color="Olive" title="b2">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c5">a9</font>  st <font color="Olive" title="b2">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  iff <font color="Olive" title="b2">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</span><br/>

<a NAME="E6:45"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<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> <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="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="yellow18.html#T9" target="_self" onmouseover="rs('yellow18/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<a NAME="E7:45"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</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="c4">b</font>,<font color="Maroon" title="c3">a</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="#E1:45"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="yellow18.html#T9" target="_self" onmouseover="rs('yellow18/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#NM4" title="ALTCAT_1:NM.4">category</a>  st <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  holds <br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Olive" title="b1">A</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b4">b</font>,<font color="Olive" title="b3">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">b9</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Olive" title="b2">B</font>  st <font color="Olive" title="b5">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b6">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b</font> holds <br/> for <font color="Olive" title="b7">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font><br/>  for <font color="Olive" title="b8">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b6">b9</font>,<font color="Olive" title="b5">a9</font>  st <font color="Olive" title="b8">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f</font> &amp; <font color="Olive" title="b7">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  holds <br/><font color="Olive" title="b8">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">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="c7">A</font>  st <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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <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="c8">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/> for <font color="Olive" title="b5">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b6">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b3">a9</font>  st <font color="Olive" title="b6">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> &amp; <font color="Olive" title="b5">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  holds <br/><font color="Olive" title="b6">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:45_1"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c7">A</font>  st <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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <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="c8">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/> for <font color="Olive" title="b5">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b6">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b3">a9</font>  st <font color="Olive" title="b6">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> &amp; <font color="Olive" title="b5">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  holds <br/><font color="Olive" title="b6">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c7">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c9">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  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="c8">B</font>  st <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Olive" title="b2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> holds <br/> for <font color="Olive" title="b3">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font><br/>  for <font color="Olive" title="b4">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b9</font>,<font color="Olive" title="b1">a9</font>  st <font color="Olive" title="b4">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> &amp; <font color="Olive" title="b3">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  holds <br/><font color="Olive" title="b4">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:45_1"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E3:45_1"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c9">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c8">B</font>  st <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Olive" title="b2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> holds <br/> for <font color="Olive" title="b3">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font><br/>  for <font color="Olive" title="b4">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b9</font>,<font color="Olive" title="b1">a9</font>  st <font color="Olive" title="b4">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> &amp; <font color="Olive" title="b3">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  holds <br/><font color="Olive" title="b4">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c11">a9</font>, <font color="Maroon" title="c12">b9</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c8">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c12">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> implies  for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font><br/>  for <font color="Olive" title="b2">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c11">a9</font>  st <font color="Olive" title="b2">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  holds <br/><font color="Olive" title="b2">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:45_1"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c11">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">and </span><br/><a NAME="E5:45_1"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c12">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font><br/>  for <font color="Olive" title="b2">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c11">a9</font>  st <font color="Olive" title="b2">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  holds <br/><font color="Olive" title="b2">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c13">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c11">a9</font>  st <font color="Olive" title="b1">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Maroon" title="c13">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  holds <br/><font color="Olive" title="b1">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c14">f9</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c12">b9</font>,<font color="Maroon" title="c11">a9</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> &amp; <font color="Maroon" title="c13">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  implies <font color="Maroon" title="c14">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</span><br/><span class="kw">assume </span><a NAME="E6:45_1"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c14">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  implies <font color="Maroon" title="c14">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</span><br/><span class="kw">assume </span><a NAME="E7:45_1"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<font color="Maroon" title="c13">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> </span><br/><a NAME="E8:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon" title="c13">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c10">b</font>
 ;<br/><a NAME="E9:45_1"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_1"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/><a NAME="E10:45_1"/>
( <font color="Maroon" title="c13">f</font> is  <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a>  &amp; <font color="Maroon" title="c13">f</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_1"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="altcat_3.html#T5" onmouseover="rs('altcat_3/T5')" onmouseout="rh()">ALTCAT_3:5</a></span>;<br/><a NAME="E11:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<font color="Maroon" title="c14">f9</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E3:45_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E4:45_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E5:45_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E6:45_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="yellow18.html#T22" target="_self" onmouseover="rs('yellow18/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E12:45_1"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c11">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E4:45_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="yellow18.html#T10" target="_self" onmouseover="rs('yellow18/T10')" onmouseout="rh()">Th10</a></span>;<br/><a NAME="E13:45_1"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c12">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E5:45_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="ref" href="yellow18.html#T10" target="_self" onmouseover="rs('yellow18/T10')" onmouseout="rh()">Th10</a></span>;<br/><a NAME="E14:45_1"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
<font color="Maroon" title="c14">f9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f9</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c11">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E3:45_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E4:45_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E5:45_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E6:45_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E9:45_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E11:45_1"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E12:45_1"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="yellow18.html#T9" target="_self" onmouseover="rs('yellow18/T9')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E15:45_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">f9</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c14">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c12">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E3:45_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E4:45_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E5:45_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E6:45_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E8:45_1"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E11:45_1"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E13:45_1"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="yellow18.html#T9" target="_self" onmouseover="rs('yellow18/T9')" onmouseout="rh()">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E16:45_1"/>
<font color="Maroon" title="c14">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:45"/>
 for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font><br/>  for <font color="Olive" title="b2">f9</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c5">a9</font>  st <font color="Olive" title="b2">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  iff <font color="Olive" title="b2">f9</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
