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<div class="add">

<span class="kw">consider </span><font color="Maroon" title="c1">F</font> being   <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E2:82_1"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c1">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">a</font>)
 <span class="kw">and </span><br/><a NAME="E3:82_1"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
 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">F<sub>1</sub></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>  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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds  <font color="Maroon" title="c1">F</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">G</font> being   <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E5:82_1"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Maroon" title="c2">G</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>)
 <span class="kw">and </span><br/><a NAME="E6:82_1"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
 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">F<sub>2</sub></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>  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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds  <font color="Maroon" title="c2">G</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E38">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c1">F</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow18.html#D2" target="_self" onmouseover="rs('yellow18/D2')" onmouseout="rh()">YELLOW18:def 2</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">  ex <font color="Olive" title="b1">G</font> being   <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Olive" title="b1">G</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">G</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="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">GF</font> = <font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">I</font> =  <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>();<br/>
<a NAME="E7:82_1"/><span class="lab"><font color="Green" title="E47">A11</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">a</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span></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; (  for <font color="Olive" title="b2">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">a</font>) holds <br/><font color="Olive" title="b2">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="82_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></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">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span></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; (  for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) holds <br/><font color="Olive" title="b1">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  ) )</span><br/>

<span class="lab"><font color="Green" title="E48">A12</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font> = 
<font color="Maroon" title="c2">G</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T33" onmouseover="rs('functor0/T33')" onmouseout="rh()">FUNCTOR0:33</a></span>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">a</font>))
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a></span>
;<br/>
<a NAME="E2:82_1_1"/><span class="lab"><font color="Green" title="E49">A13</font></span>: 
<span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T29" onmouseover="rs('functor0/T29')" onmouseout="rh()">FUNCTOR0:29</a></span>;<br/>
<a NAME="E3:82_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A14</font></span>: 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span></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="#E3:82"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E1:82_1_1"><span class="lab"><font color="Green" title="E48">A12</font></span></a></span>;<br/>
<a NAME="E4:82_1_1"/><span class="lab"><font color="Green" title="E51">A15</font></span>: 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span></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="#E3:82"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E1:82_1_1"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E49">A13</font></span></a></span>;<br/>
<a NAME="E5:82_1_1"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E1:82_1_1"><span class="lab"><font color="Green" title="E48">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:82_1_1"/>
( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span></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; (  for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) holds <br/><font color="Olive" title="b1">f</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="#E3:82_1_1"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="txt" href="#E4:82_1_1"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="ref" href="yellow18.html#T41" target="_self" onmouseover="rs('yellow18/T41')" onmouseout="rh()">Th41</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>
<a NAME="E8:82_1"/><span class="lab"><font color="Green" title="E48">A16</font></span>: 
 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">F<sub>1</sub></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>  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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b4">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>  st <font color="Olive" title="b4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">a</font>) holds <br/> for <font color="Olive" title="b5">g2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span>  st <font color="Olive" title="b5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b2">b</font>) holds <br/><font color="Olive" title="b5">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b4">g1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="82_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></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="c5">a</font>,<font color="Maroon" title="c6">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>  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="c5">a</font>,<font color="Maroon" title="c6">b</font><br/>  for <font color="Olive" title="b2">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>  st <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) holds <br/> for <font color="Olive" title="b3">g2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span>  st <font color="Olive" title="b3">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">b</font>) holds <br/><font color="Olive" title="b3">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b2">g1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:82_1_2"/><span class="lab"><font color="Green" title="E49">A17</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">a</font>,<font color="Maroon" title="c6">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> 
 ; <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="c5">a</font>,<font color="Maroon" title="c6">b</font><br/>  for <font color="Olive" title="b2">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>  st <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) holds <br/> for <font color="Olive" title="b3">g2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span>  st <font color="Olive" title="b3">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">b</font>) holds <br/><font color="Olive" title="b3">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b2">g1</font></span><br/>

<a NAME="E2:82_1_2"/><span class="lab"><font color="Green" title="E50">A18</font></span>: 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span></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="#E7:82_1"><span class="lab"><font color="Green" title="E47">A11</font></span></a></span>;<br/>
<a NAME="E3:82_1_2"/><span class="lab"><font color="Green" title="E51">A19</font></span>: 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">b</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span></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="#E7:82_1"><span class="lab"><font color="Green" title="E47">A11</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">g2</font> = <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">b</font>) as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:82_1"><span class="lab"><font color="Green" title="E47">A11</font></span></a></span>;<br/>
<a NAME="E5:82_1_2"/><span class="lab"><font color="Green" title="E52">A20</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span></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">by</span> <span class="lab"><a class="txt" href="#E1:82_1_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="functor0.html#D18" onmouseover="rs('functor0/D18')" onmouseout="rh()">FUNCTOR0:def 18</a></span>;<br/>
<a NAME="E6:82_1_2"/><span class="lab"><font color="Green" title="E53">A21</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span></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">by</span> <span class="lab"><a class="txt" href="#E1:82_1_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="functor0.html#D18" onmouseover="rs('functor0/D18')" onmouseout="rh()">FUNCTOR0:def 18</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) holds <br/> for <font color="Olive" title="b2">g2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span>  st <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">b</font>) holds <br/><font color="Olive" title="b2">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">g1</font></span><br/>

<a NAME="E7:82_1_2"/><span class="lab"><font color="Green" title="E54">A22</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c8">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="functor3.html#T6" onmouseover="rs('functor3/T6')" onmouseout="rh()">FUNCTOR3:6</a></span>;<br/>
<a NAME="E8:82_1_2"/><span class="lab"><font color="Green" title="E55">A23</font></span>: 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a></span>;<br/>
<a NAME="E9:82_1_2"/><span class="lab"><font color="Green" title="E56">A24</font></span>: 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a></span>;<br/>
<a NAME="E10:82_1_2"/><span class="lab"><font color="Green" title="E57">A25</font></span>: 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E1:82_1_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>;<br/>
<a NAME="E11:82_1_2"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span></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">by</span> <span class="lab"><a class="txt" href="#E1:82_1_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="functor0.html#D18" onmouseover="rs('functor0/D18')" onmouseout="rh()">FUNCTOR0:def 18</a></span>;<br/>
<a NAME="E12:82_1_2"/><span class="kw">then </span>
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">b</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:82_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E7:82_1_2"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="txt" href="#E8:82_1_2"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="txt" href="#E9:82_1_2"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="txt" href="#E10:82_1_2"><span class="lab"><font color="Green" title="E57">A25</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c7">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c8">f</font></span>)</span> = 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">b</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">f</font>))
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1_2"><span class="lab"><font color="Green" title="E51">A19</font></span></a>, <a class="txt" href="#E5:82_1_2"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="ref" href="yellow18.html#T36" target="_self" onmouseover="rs('yellow18/T36')" onmouseout="rh()">Th36</a></span>
<br/>.= 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E1:82_1_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="functor0.html#T31" onmouseover="rs('functor0/T31')" onmouseout="rh()">FUNCTOR0:31</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:82_1_2"/>
 for <font color="Olive" title="b1">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">a</font>) holds <br/> for <font color="Olive" title="b2">g2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c6">b</font></span>)</span>  st <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">b</font>) holds <br/><font color="Olive" title="b2">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_2"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="txt" href="#E6:82_1_2"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="yellow18.html#T36" target="_self" onmouseover="rs('yellow18/T36')" onmouseout="rh()">Th36</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>
<a NAME="E9:82_1"/>
 ex <font color="Olive" title="b1">t</font> being    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; (  for <font color="Olive" title="b2">a</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Olive" title="b1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b2">a</font>) ) )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="yellow18.html#S15" target="_self" onmouseover="rs('yellow18/S15')" onmouseout="rh()">YELLOW18:sch 15</a>(<a class="txt" href="#E7:82_1"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="txt" href="#E8:82_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E10:82_1"/>
<font color="Maroon" title="c2">G</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c1">F</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>1</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</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">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">I</font> =  <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>();<br/>
<span class="kw">set </span><font color="Maroon" title="c6">FG</font> = <font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font>;<br/>
<a NAME="E11:82_1"/><span class="lab"><font color="Green" title="E49">A26</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>2</sub></font>() holds <br/> ( <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b1">a</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span></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; (  for <font color="Olive" title="b2">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b1">a</font>) holds <br/><font color="Olive" title="b2">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="82_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>2</sub></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">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span></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; (  for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) holds <br/><font color="Olive" title="b1">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  ) )</span><br/>

<span class="lab"><font color="Green" title="E50">A27</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font> = 
<font color="Maroon" title="c1">F</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T33" onmouseover="rs('functor0/T33')" onmouseout="rh()">FUNCTOR0:33</a></span>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">a</font>))
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a></span>
;<br/>
<a NAME="E2:82_1_3"/><span class="lab"><font color="Green" title="E51">A28</font></span>: 
<span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T29" onmouseover="rs('functor0/T29')" onmouseout="rh()">FUNCTOR0:29</a></span>;<br/>
<a NAME="E3:82_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A29</font></span>: 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span></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="#E4:82"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:82_1_3"><span class="lab"><font color="Green" title="E50">A27</font></span></a></span>;<br/>
<a NAME="E4:82_1_3"/><span class="lab"><font color="Green" title="E53">A30</font></span>: 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span></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="#E4:82"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:82_1_3"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E2:82_1_3"><span class="lab"><font color="Green" title="E51">A28</font></span></a></span>;<br/>
<a NAME="E5:82_1_3"/>
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:82_1_3"><span class="lab"><font color="Green" title="E50">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:82_1_3"/>
( <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span></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; (  for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) holds <br/><font color="Olive" title="b1">f</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="#E3:82_1_3"><span class="lab"><font color="Green" title="E52">A29</font></span></a>, <a class="txt" href="#E4:82_1_3"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="ref" href="yellow18.html#T41" target="_self" onmouseover="rs('yellow18/T41')" onmouseout="rh()">Th41</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>
<a NAME="E12:82_1"/><span class="lab"><font color="Green" title="E50">A31</font></span>: 
 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">F<sub>2</sub></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>  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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b4">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>  st <font color="Olive" title="b4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b1">a</font>) holds <br/> for <font color="Olive" title="b5">g2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span>  st <font color="Olive" title="b5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b2">b</font>) holds <br/><font color="Olive" title="b5">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b4">g1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="82_1_4"><span class="kw">proof </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#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>2</sub></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="c7">a</font>,<font color="Maroon" title="c8">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>  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="c7">a</font>,<font color="Maroon" title="c8">b</font><br/>  for <font color="Olive" title="b2">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>  st <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) holds <br/> for <font color="Olive" title="b3">g2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span>  st <font color="Olive" title="b3">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c8">b</font>) holds <br/><font color="Olive" title="b3">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b2">g1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:82_1_4"/><span class="lab"><font color="Green" title="E51">A32</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="c7">a</font>,<font color="Maroon" title="c8">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> 
 ; <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="c7">a</font>,<font color="Maroon" title="c8">b</font><br/>  for <font color="Olive" title="b2">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>  st <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) holds <br/> for <font color="Olive" title="b3">g2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span>  st <font color="Olive" title="b3">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c8">b</font>) holds <br/><font color="Olive" title="b3">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b2">g1</font></span><br/>

<a NAME="E2:82_1_4"/><span class="lab"><font color="Green" title="E52">A33</font></span>: 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span></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="#E11:82_1"><span class="lab"><font color="Green" title="E49">A26</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">g1</font> = <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:82_1"><span class="lab"><font color="Green" title="E49">A26</font></span></a></span>;<br/>
<a NAME="E4:82_1_4"/><span class="lab"><font color="Green" title="E53">A34</font></span>: 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c8">b</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span></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="#E11:82_1"><span class="lab"><font color="Green" title="E49">A26</font></span></a></span>;<br/>
<a NAME="E5:82_1_4"/><span class="lab"><font color="Green" title="E54">A35</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span></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">by</span> <span class="lab"><a class="txt" href="#E1:82_1_4"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="ref" href="functor0.html#D18" onmouseover="rs('functor0/D18')" onmouseout="rh()">FUNCTOR0:def 18</a></span>;<br/>
<a NAME="E6:82_1_4"/><span class="lab"><font color="Green" title="E55">A36</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span></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">by</span> <span class="lab"><a class="txt" href="#E1:82_1_4"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="ref" href="functor0.html#D18" onmouseover="rs('functor0/D18')" onmouseout="rh()">FUNCTOR0:def 18</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">a</font>,<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">  for <font color="Olive" title="b1">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) holds <br/> for <font color="Olive" title="b2">g2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span>  st <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c8">b</font>) holds <br/><font color="Olive" title="b2">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">g1</font></span><br/>

<a NAME="E7:82_1_4"/><span class="lab"><font color="Green" title="E56">A37</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c10">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_4"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="ref" href="functor3.html#T6" onmouseover="rs('functor3/T6')" onmouseout="rh()">FUNCTOR3:6</a></span>;<br/>
<a NAME="E8:82_1_4"/><span class="lab"><font color="Green" title="E57">A38</font></span>: 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a></span>;<br/>
<a NAME="E9:82_1_4"/><span class="lab"><font color="Green" title="E58">A39</font></span>: 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a></span>;<br/>
<a NAME="E10:82_1_4"/><span class="lab"><font color="Green" title="E59">A40</font></span>: 
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c10">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:82_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E1:82_1_4"><span class="lab"><font color="Green" title="E51">A32</font></span></a></span>;<br/>
<a NAME="E11:82_1_4"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span></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">by</span> <span class="lab"><a class="txt" href="#E1:82_1_4"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="ref" href="functor0.html#D18" onmouseover="rs('functor0/D18')" onmouseout="rh()">FUNCTOR0:def 18</a></span>;<br/>
<a NAME="E12:82_1_4"/><span class="kw">then </span>
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">a</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">b</font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c10">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E7:82_1_4"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="txt" href="#E8:82_1_4"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E9:82_1_4"><span class="lab"><font color="Green" title="E58">A39</font></span></a>, <a class="txt" href="#E10:82_1_4"><span class="lab"><font color="Green" title="E59">A40</font></span></a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c9">g1</font> = 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c7">a</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">b</font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c10">f</font>)) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_4"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E5:82_1_4"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="ref" href="yellow18.html#T36" target="_self" onmouseover="rs('yellow18/T36')" onmouseout="rh()">Th36</a></span>
<br/>.= 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c8">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:82"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E1:82_1_4"><span class="lab"><font color="Green" title="E51">A32</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c8">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c10">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_4"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="ref" href="functor0.html#T31" onmouseover="rs('functor0/T31')" onmouseout="rh()">FUNCTOR0:31</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:82_1_4"/>
 for <font color="Olive" title="b1">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c7">a</font></span>)</span>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c7">a</font>) holds <br/> for <font color="Olive" title="b2">g2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K2" title="FUNCTOR0:func.2">.</a> <font color="Maroon" title="c8">b</font></span>)</span>  st <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c8">b</font>) holds <br/><font color="Olive" title="b2">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="functor0.html#K5" title="FUNCTOR0:func.5">.</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_4"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E6:82_1_4"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="ref" href="yellow18.html#T36" target="_self" onmouseover="rs('yellow18/T36')" onmouseout="rh()">Th36</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>
<a NAME="E13:82_1"/>
 ex <font color="Olive" title="b1">t</font> being    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of  <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font> st <br/>(  <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; (  for <font color="Olive" title="b2">a</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Olive" title="b1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b2">a</font>) ) )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="yellow18.html#S15" target="_self" onmouseover="rs('yellow18/S15')" onmouseout="rh()">YELLOW18:sch 15</a>(<a class="txt" href="#E11:82_1"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="txt" href="#E12:82_1"><span class="lab"><font color="Green" title="E50">A31</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E14:82_1"/>
<font color="Maroon" title="c1">F</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c2">G</font>, <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon">F<sub>2</sub></font>() <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</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"> verum</span><br/>


</div>
