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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> holds <br/>for <font color="Olive" title="b3">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> holds <br/> (  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Olive" title="b1">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Olive" title="b2">F2</font> &amp;  ex <font color="Olive" title="b4">G</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Olive" title="b1">F1</font>, <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Olive" title="b2">F2</font> st <br/>for <font color="Olive" title="b5">s</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b3">t</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b5">s</font> holds <br/>for <font color="Olive" title="b6">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Olive" title="b4">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Olive" title="b1">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b6">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Olive" title="b2">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b6">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Olive" title="b5">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b6">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/><b>let </b><font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c5">F2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F2</font> implies for <font color="Olive" title="b1">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> holds <br/> (  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font> &amp;  ex <font color="Olive" title="b2">G</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font>, <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font> st <br/>for <font color="Olive" title="b3">s</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">t</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b3">s</font> holds <br/>for <font color="Olive" title="b4">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Olive" title="b2">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b4">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b4">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Olive" title="b3">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b4">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>







<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> holds <br/> (  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font> &amp;  ex <font color="Olive" title="b2">G</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font>, <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font> st <br/>for <font color="Olive" title="b3">s</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">t</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b3">s</font> holds <br/>for <font color="Olive" title="b4">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Olive" title="b2">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b4">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b4">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Olive" title="b3">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b4">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<a NAME="E2:29"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c5">F2</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">t</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font> &amp;  ex <font color="Olive" title="b1">G</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font>, <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font> st <br/>for <font color="Olive" title="b2">s</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C</font>  st <font color="Maroon" title="c6">t</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">s</font> holds <br/>for <font color="Olive" title="b3">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Olive" title="b1">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b3">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b3">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b3">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<a NAME="E3:29"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="cat_2.html#T33" title="CAT_2:th.33">CAT_2:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c6">t</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C</font> ;<br/>
<div><i><font color="Green" title="E28">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S1</font>, <font color="Olive" title="b2">S2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Olive" title="b2">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> holds <br/>for <font color="Olive" title="b3">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b3">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b3">b</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c9">S1</font>, <font color="Maroon" title="c10">S2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c10">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> implies for <font color="Olive" title="b1">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">b</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:29_1"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c9">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c10">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">b</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c11">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font></span>)</span></span><br/><i><font color="Green" title="E30">A5</font></i>: <font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font>
<b>by </b><i><a class="txt" href="#E1:29_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="isocat_2.html#T24" target="_self" title="ISOCAT_2:th.24">Th24</a></i>
<br/>.= 
<font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c11">b</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>
<b>by </b><i><a class="ref" href="isocat_2.html#T12" target="_self" title="ISOCAT_2:th.12">Th12</a></i>
;<br/><i><font color="Green" title="E31">A6</font></i>: <font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font>
<b>by </b><i><a class="txt" href="#E1:29_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="isocat_2.html#T24" target="_self" title="ISOCAT_2:th.24">Th24</a></i>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c11">b</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>
<b>by </b><i><a class="ref" href="isocat_2.html#T12" target="_self" title="ISOCAT_2:th.12">Th12</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font> = 
<font color="Maroon" title="c7">s</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c11">b</font>
<b>by </b><i><a class="ref" href="cat_2.html#T3" title="CAT_2:th.3">CAT_2:3</a></i>
<br/>.= 
<font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c11">b</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>
<b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E5:29_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="#E2:29_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="#E3:29_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="isocat_2.html#E26"><i><font color="Green" title="E21">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E29">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S1</font>, <font color="Olive" title="b2">S2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Olive" title="b2">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> holds <br/><font color="Olive" title="b1">S1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Olive" title="b2">S2</font></span><br/><b>let </b><font color="Maroon" title="c9">S1</font>, <font color="Maroon" title="c10">S2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c10">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> implies <font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c10">S2</font> )</span><br/><b>assume </b><a NAME="E1:29_2"/>
( <font color="Maroon" title="c9">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c10">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c10">S2</font></span><br/><a NAME="E2:29_2"/><b>then </b>
for <font color="Olive" title="b1">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>  holds  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">b</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E5:29"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:29_2"/>
<font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c10">S2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E30">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S1</font>, <font color="Olive" title="b2">S2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Olive" title="b2">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> holds <br/><span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font> is    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font></span><br/><b>let </b><font color="Maroon" title="c9">S1</font>, <font color="Maroon" title="c10">S2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c10">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> implies <span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font> is    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c9">S1</font>,<font color="Maroon" title="c10">S2</font> )</span><br/><b>assume </b><a NAME="E1:29_3"/><i><font color="Green" title="E31">A9</font></i>: 
( <font color="Maroon" title="c9">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c10">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font> is    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c9">S1</font>,<font color="Maroon" title="c10">S2</font></span><br/><b>thus </b><a NAME="E2:29_3"/>
<span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font> is    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c9">S1</font>,<font color="Maroon" title="c10">S2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_3_1"/>
<font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c10">S2</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="#E1:29_3"><i><font color="Green" title="E31">A9</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D3" title="NATTRA_1:def.3">NATTRA_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive">b<sub>1</sub></font></span><br/>

<b>let </b><font color="Maroon" title="c11">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font></span><br/>

<a NAME="E2:29_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:29"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="#E1:29_3"><i><font color="Green" title="E31">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:29_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green" title="E31">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S1</font>, <font color="Olive" title="b2">S2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font><br/> for <font color="Olive" title="b3">T</font> being   <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>  st <font color="Olive" title="b1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Olive" title="b2">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Olive" title="b3">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font> holds <br/>for <font color="Olive" title="b4">b1</font>, <font color="Olive" title="b5">b2</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b4">b1</font>,<font color="Olive" title="b5">b2</font> <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="b6">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b4">b1</font>,<font color="Olive" title="b5">b2</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b5">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b6">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b4">b1</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c9">S1</font>, <font color="Maroon" title="c10">S2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being   <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c9">S1</font>,<font color="Maroon" title="c10">S2</font>  st <font color="Maroon" title="c9">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c10">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Olive" title="b1">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font> holds <br/>for <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font> <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="b4">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b3">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b4">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b2">b1</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c11">T</font> be    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c9">S1</font>,<font color="Maroon" title="c10">S2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c10">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c11">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font> implies for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font> <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="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b2">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</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"><font color="Maroon" title="c10">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b1">b1</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:29_4"/><i><font color="Green" title="E32">A11</font></i>: 
( <font color="Maroon" title="c9">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c10">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c11">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c8">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font> <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="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b2">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</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"><font color="Maroon" title="c10">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b1">b1</font></span>)</span></span><br/><a NAME="E2:29_4"/><b>then </b><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c10">S2</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c12">b1</font>, <font color="Maroon" title="c13">b2</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c12">b1</font>,<font color="Maroon" title="c13">b2</font> <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="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c12">b1</font>,<font color="Maroon" title="c13">b2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c13">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</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"><font color="Maroon" title="c10">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c12">b1</font></span>)</span> )</span><br/><b>assume </b><a NAME="E3:29_4"/><i><font color="Green" title="E34">A13</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c12">b1</font>,<font color="Maroon" title="c13">b2</font> <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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c12">b1</font>,<font color="Maroon" title="c13">b2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c13">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</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"><font color="Maroon" title="c10">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c12">b1</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c14">f</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c12">b1</font>,<font color="Maroon" title="c13">b2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c13">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c12">b1</font></span>)</span></span><br/><a NAME="E4:29_4"/><i><font color="Green" title="E35">A14</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">g</font> = <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c14">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c13">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> <b>by </b><i><a class="txt" href="#E3:29_4"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="cat_2.html#T43" title="CAT_2:th.43">CAT_2:43</a></i>;<br/><a NAME="E6:29_4"/><i><font color="Green" title="E36">A15</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c13">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</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> 
 <b>by </b><i><a class="txt" href="#E3:29_4"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="#E4:29_4"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="isocat_2.html#T13" target="_self" title="ISOCAT_2:th.13">Th13</a></i>;<br/><a NAME="E7:29_4"/><b>then </b><i><font color="Green" title="E37">A16</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c13">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</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> 
 <b>by </b><i><a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/><a NAME="E8:29_4"/><i><font color="Green" title="E38">A17</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c13">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</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> 
 <b>by </b><i><a class="txt" href="#E6:29_4"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/><a NAME="E9:29_4"/><i><font color="Green" title="E39">A18</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</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> 
 <b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/><a NAME="E10:29_4"/><i><font color="Green" title="E40">A19</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c13">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c13">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</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> 
 <b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/><a NAME="E11:29_4"/><i><font color="Green" title="E41">A20</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c12">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b2</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E3:29_4"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/><a NAME="E12:29_4"/><i><font color="Green" title="E42">A21</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c12">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b2</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E3:29_4"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/><a NAME="E13:29_4"/><i><font color="Green" title="E43">A22</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c12">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c12">b1</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E2:29_4"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/><a NAME="E14:29_4"/><i><font color="Green" title="E44">A23</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b2</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E2:29_4"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/><i><font color="Green" title="E45">A24</font></i>: <font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c12">b1</font> = 
<font color="Maroon" title="c11">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b1</font>
<b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="#E1:29_4"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c7">s</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">b1</font>
<b>by </b><i><a class="txt" href="#E1:29_4"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="cat_2.html#T3" title="CAT_2:th.3">CAT_2:3</a></i>
<br/>.= 
<font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>
<b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
;<br/><i><font color="Green" title="E46">A25</font></i>: <font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c13">b2</font> = 
<font color="Maroon" title="c11">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b2</font>
<b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="#E1:29_4"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c7">s</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c13">b2</font>
<b>by </b><i><a class="txt" href="#E1:29_4"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="cat_2.html#T3" title="CAT_2:th.3">CAT_2:3</a></i>
<br/>.= 
<font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c13">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>
<b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
;<br/><i><font color="Green" title="E47">A26</font></i>: <font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">f</font>
<b>by </b><i><a class="txt" href="#E1:29_4"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="isocat_2.html#T24" target="_self" title="ISOCAT_2:th.24">Th24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f</font>
<b>by </b><i><a class="txt" href="#E3:29_4"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">F1</font> <a href="cat_2.html#K14" title="CAT_2:func.14">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c14">f</font>
<b>by </b><i><a class="ref" href="cat_2.html#T47" title="CAT_2:th.47">CAT_2:47</a></i>
<br/>.= 
<font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c15">g</font>
<b>by </b><i><a class="txt" href="#E6:29_4"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
;<br/><i><font color="Green" title="E48">A27</font></i>: <font color="Maroon" title="c10">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">f</font>
<b>by </b><i><a class="txt" href="#E1:29_4"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="isocat_2.html#T24" target="_self" title="ISOCAT_2:th.24">Th24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f</font>
<b>by </b><i><a class="txt" href="#E3:29_4"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="cat_2.html#K14" title="CAT_2:func.14">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c8">a</font></span>)</span>,<font color="Maroon" title="c14">f</font>
<b>by </b><i><a class="ref" href="cat_2.html#T47" title="CAT_2:th.47">CAT_2:47</a></i>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c15">g</font>
<b>by </b><i><a class="txt" href="#E6:29_4"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c13">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c13">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c15">g</font></span>)</span>
<b>by </b><i><a class="txt" href="#E11:29_4"><i><font color="Green" title="E41">A20</font></i></a>, <a class="txt" href="#E14:29_4"><i><font color="Green" title="E44">A23</font></i></a>, <a class="txt" href="#E16:29_4"><i><font color="Green" title="E46">A25</font></i></a>, <a class="txt" href="#E17:29_4"><i><font color="Green" title="E47">A26</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c13">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c15">g</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:29_4"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="#E10:29_4"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E6:29_4"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c12">b1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E8:29_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="#E9:29_4"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="#E15:29_4"><i><font color="Green" title="E45">A24</font></i></a>, <a class="txt" href="#E18:29_4"><i><font color="Green" title="E48">A27</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c12">b1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E12:29_4"><i><font color="Green" title="E42">A21</font></i></a>, <a class="txt" href="#E13:29_4"><i><font color="Green" title="E43">A22</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">s</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C</font>  st <font color="Maroon" title="c6">t</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">s</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<div><i><font color="Green" title="E32">A28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> implies  ex <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">m</font>,<font color="Olive" title="b1">o</font>] ) )</span><br/><b>assume </b><a NAME="E1:29_5"/>
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">m</font>,<font color="Olive" title="b1">o</font>] )</span><br/><b>then reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">m</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> ;<br/><b>take </b><font color="Maroon" title="c10">o</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c10">o</font>] )</span><br/><b>reconsider </b><font color="Maroon" title="c11">S1</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c12">S2</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <b>by </b><i><a class="ref" href="isocat_2.html#T25" target="_self" title="ISOCAT_2:th.25">Th25</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">T</font> = <span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c9">a</font> as    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c11">S1</font>,<font color="Maroon" title="c12">S2</font> <b>by </b><i><a class="txt" href="#E7:29"><i><font color="Green" title="E30">A8</font></i></a></i>;<br/><a NAME="E5:29_5"/><i><font color="Green" title="E33">A29</font></i>: 
<font color="Maroon" title="c11">S1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c12">S2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_5_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_5_1"/>
<font color="Maroon" title="c11">S1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c12">S2</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E29">A7</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c11">S1</font>,<font color="Maroon" title="c12">S2</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c13">T</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E2:29_5_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E8:29"><i><font color="Green" title="E31">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E6:29_5"/>
<font color="Maroon" title="c13">T</font> is    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c11">S1</font>,<font color="Maroon" title="c12">S2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_5_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_5_2"/>
<font color="Maroon" title="c11">S1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c12">S2</font>
 <b>by </b><i><a class="txt" href="#E5:29_5"><i><font color="Green" title="E33">A29</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E2:29_5_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E8:29"><i><font color="Green" title="E31">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E7:29_5"/><b>then </b>
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="#E5:29_5"><i><font color="Green" title="E33">A29</font></i></a>, <a class="ref" href="nattra_1.html#T35" title="NATTRA_1:th.35">NATTRA_1:35</a></i>;<br/><b>hence </b><a NAME="E8:29_5"/>
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c10">o</font>]</span><br/><b>thus </b><a NAME="E9:29_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c10">o</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span><b> such that </b><br/><a NAME="E11:29"/><i><font color="Green" title="E33">A30</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E9:29"><i><font color="Green" title="E32">A28</font></i></a>);<br/></i>
<div><i><font color="Green" title="E34">A31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c10">S1</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c11">S2</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <b>by </b><i><a class="ref" href="isocat_2.html#T25" target="_self" title="ISOCAT_2:th.25">Th25</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">T</font> = <span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c9">a</font> as    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c10">S1</font>,<font color="Maroon" title="c11">S2</font> <b>by </b><i><a class="txt" href="#E7:29"><i><font color="Green" title="E30">A8</font></i></a></i>;<br/><a NAME="E3:29_6"/><i><font color="Green" title="E35">A32</font></i>: 
<font color="Maroon" title="c10">S1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">S2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_6_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_6_1"/>
<font color="Maroon" title="c10">S1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c11">S2</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E29">A7</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c10">S1</font>,<font color="Maroon" title="c11">S2</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c12">T</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E2:29_6_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E8:29"><i><font color="Green" title="E31">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E4:29_6"/><i><font color="Green" title="E36">A33</font></i>: 
<font color="Maroon" title="c12">T</font> is    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">S1</font>,<font color="Maroon" title="c11">S2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_6_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_6_2"/>
<font color="Maroon" title="c10">S1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">S2</font>
 <b>by </b><i><a class="txt" href="#E3:29_6"><i><font color="Green" title="E35">A32</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E2:29_6_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">S1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">T</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E8:29"><i><font color="Green" title="E31">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E5:29_6"/>
<font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">S1</font>,<font color="Maroon" title="c11">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E11:29"><i><font color="Green" title="E33">A30</font></i></a></i>;<br/><a NAME="E6:29_6"/><b>then </b>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">S1</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">S2</font> )
 <b>by </b><i><a class="txt" href="#E4:29_6"><i><font color="Green" title="E36">A33</font></i></a>, <a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>;<br/><b>hence </b><a NAME="E7:29_6"/>
<font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E13:29"/><b>then </b><i><font color="Green" title="E35">A34</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>  holds  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">a</font></span>)</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> 
 
;<br/>
<a NAME="E14:29"/><b>then </b><i><font color="Green" title="E36">A35</font></i>: 
 <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E15:29"/>
<font color="Maroon" title="c8">G</font> is    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font>, <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_7"/>
 <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font>
 <b>by </b><i><a class="txt" href="#E13:29"><i><font color="Green" title="E35">A34</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D3" title="NATTRA_1:def.3">NATTRA_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive">b<sub>1</sub></font></span><br/>

<b>let </b><font color="Maroon" title="c9">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font>,<span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font></span><br/>

<a NAME="E2:29_7"/>
<font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E12:29"><i><font color="Green" title="E34">A31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:29_7"/>
<font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font>,<span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">G</font> = <font color="Maroon" title="c8">G</font> as    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font>, <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font> ;<br/>
<div><i><font color="Green" title="E37">A36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <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="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</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="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:29_8"/><i><font color="Green" title="E38">A37</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</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="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c12">f</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">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="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c12">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:29_8"/><i><font color="Green" title="E39">A38</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font>
 <b>by </b><i><a class="ref" href="isocat_2.html#T24" target="_self" title="ISOCAT_2:th.24">Th24</a></i>;<br/><a NAME="E4:29_8"/><b>then </b><i><font color="Green" title="E40">A39</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font>
 <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/><a NAME="E5:29_8"/><i><font color="Green" title="E41">A40</font></i>: 
<font color="Maroon" title="c5">F2</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="ref" href="isocat_2.html#T24" target="_self" title="ISOCAT_2:th.24">Th24</a></i>;<br/><a NAME="E6:29_8"/><b>then </b><i><font color="Green" title="E42">A41</font></i>: 
<font color="Maroon" title="c5">F2</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/><a NAME="E7:29_8"/><i><font color="Green" title="E43">A42</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="ref" href="isocat_2.html#T24" target="_self" title="ISOCAT_2:th.24">Th24</a></i>;<br/><a NAME="E8:29_8"/><b>then </b><i><font color="Green" title="E44">A43</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/><a NAME="E9:29_8"/><i><font color="Green" title="E45">A44</font></i>: 
<font color="Maroon" title="c5">F2</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font>
 <b>by </b><i><a class="ref" href="isocat_2.html#T24" target="_self" title="ISOCAT_2:th.24">Th24</a></i>;<br/><a NAME="E10:29_8"/><b>then </b><i><font color="Green" title="E46">A45</font></i>: 
<font color="Maroon" title="c5">F2</font> <a href="cat_2.html#K15" title="CAT_2:func.15">?-</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font>
 <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">S1</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c15">S2</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c16">S3</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font>, <font color="Maroon" title="c17">S4</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <b>by </b><i><a class="ref" href="isocat_2.html#T25" target="_self" title="ISOCAT_2:th.25">Th25</a></i>;<br/><a NAME="E12:29_8"/><i><font color="Green" title="E47">A46</font></i>: 
<font color="Maroon" title="c14">S1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c15">S2</font>
 <b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E5:29_8"><i><font color="Green" title="E41">A40</font></i></a>, <a class="txt" href="#E7:29_8"><i><font color="Green" title="E43">A42</font></i></a>, <a class="ref" href="isocat_2.html#T15" target="_self" title="ISOCAT_2:th.15">Th15</a></i>;<br/><a NAME="E13:29_8"/><b>then </b><i><font color="Green" title="E48">A47</font></i>: 
<font color="Maroon" title="c14">S1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c15">S2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/><a NAME="E14:29_8"/><i><font color="Green" title="E49">A48</font></i>: 
<font color="Maroon" title="c15">S2</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c17">S4</font>
 <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="txt" href="#E5:29_8"><i><font color="Green" title="E41">A40</font></i></a>, <a class="txt" href="#E9:29_8"><i><font color="Green" title="E45">A44</font></i></a>, <a class="ref" href="isocat_2.html#T17" target="_self" title="ISOCAT_2:th.17">Th17</a></i>;<br/><a NAME="E15:29_8"/><b>then </b><i><font color="Green" title="E50">A49</font></i>: 
<font color="Maroon" title="c15">S2</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c17">S4</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/><a NAME="E16:29_8"/><i><font color="Green" title="E51">A50</font></i>: 
<font color="Maroon" title="c14">S1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c16">S3</font>
 <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="txt" href="#E3:29_8"><i><font color="Green" title="E39">A38</font></i></a>, <a class="txt" href="#E7:29_8"><i><font color="Green" title="E43">A42</font></i></a>, <a class="ref" href="isocat_2.html#T17" target="_self" title="ISOCAT_2:th.17">Th17</a></i>;<br/><a NAME="E17:29_8"/><b>then </b><i><font color="Green" title="E52">A51</font></i>: 
<font color="Maroon" title="c14">S1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c16">S3</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/><a NAME="E18:29_8"/><i><font color="Green" title="E53">A52</font></i>: 
<font color="Maroon" title="c16">S3</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c17">S4</font>
 <b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E3:29_8"><i><font color="Green" title="E39">A38</font></i></a>, <a class="txt" href="#E9:29_8"><i><font color="Green" title="E45">A44</font></i></a>, <a class="ref" href="isocat_2.html#T15" target="_self" title="ISOCAT_2:th.15">Th15</a></i>;<br/><a NAME="E19:29_8"/><b>then </b><i><font color="Green" title="E54">A53</font></i>: 
<font color="Maroon" title="c16">S3</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c17">S4</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/><a NAME="E20:29_8"/><i><font color="Green" title="E55">A54</font></i>: 
<font color="Maroon" title="c14">S1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c17">S4</font>
 <b>by </b><i><a class="txt" href="#E13:29_8"><i><font color="Green" title="E48">A47</font></i></a>, <a class="txt" href="#E15:29_8"><i><font color="Green" title="E50">A49</font></i></a>, <a class="ref" href="nattra_1.html#T19" title="NATTRA_1:th.19">NATTRA_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">T13</font> = <font color="Maroon" title="c4">F1</font> <a href="isocat_2.html#K4" title="ISOCAT_2:func.4">?-</a> <font color="Maroon" title="c12">f</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">S1</font>,<font color="Maroon" title="c16">S3</font> <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="txt" href="#E4:29_8"><i><font color="Green" title="E40">A39</font></i></a>, <a class="txt" href="#E7:29_8"><i><font color="Green" title="E43">A42</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">T12</font> = <span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c10">a</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">S1</font>,<font color="Maroon" title="c15">S2</font> <b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E5:29_8"><i><font color="Green" title="E41">A40</font></i></a>, <a class="txt" href="#E7:29_8"><i><font color="Green" title="E43">A42</font></i></a>, <a class="ref" href="isocat_2.html#T15" target="_self" title="ISOCAT_2:th.15">Th15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">T24</font> = <font color="Maroon" title="c5">F2</font> <a href="isocat_2.html#K4" title="ISOCAT_2:func.4">?-</a> <font color="Maroon" title="c12">f</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c15">S2</font>,<font color="Maroon" title="c17">S4</font> <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="txt" href="#E5:29_8"><i><font color="Green" title="E41">A40</font></i></a>, <a class="txt" href="#E10:29_8"><i><font color="Green" title="E46">A45</font></i></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">T34</font> = <span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c11">b</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c16">S3</font>,<font color="Maroon" title="c17">S4</font> <b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E3:29_8"><i><font color="Green" title="E39">A38</font></i></a>, <a class="txt" href="#E9:29_8"><i><font color="Green" title="E45">A44</font></i></a>, <a class="ref" href="isocat_2.html#T15" target="_self" title="ISOCAT_2:th.15">Th15</a></i>;<br/><a NAME="E25:29_8"/><i><font color="Green" title="E56">A55</font></i>: 
<span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">S1</font>,<font color="Maroon" title="c16">S3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">T13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E4:29_8"><i><font color="Green" title="E40">A39</font></i></a>, <a class="txt" href="#E8:29_8"><i><font color="Green" title="E44">A43</font></i></a>, <a class="ref" href="isocat_2.html#D4" target="_self" title="ISOCAT_2:def.4">Def4</a></i>;<br/><a NAME="E26:29_8"/><i><font color="Green" title="E57">A56</font></i>: 
<span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">S2</font>,<font color="Maroon" title="c17">S4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">T24</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E6:29_8"><i><font color="Green" title="E42">A41</font></i></a>, <a class="txt" href="#E10:29_8"><i><font color="Green" title="E46">A45</font></i></a>, <a class="ref" href="isocat_2.html#D4" target="_self" title="ISOCAT_2:def.4">Def4</a></i>;<br/><i><font color="Green" title="E58">A57</font></i>: <font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">b</font> = 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font>
<b>by </b><i><a class="txt" href="#E14:29"><i><font color="Green" title="E36">A35</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">S3</font>,<font color="Maroon" title="c17">S4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c21">T34</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E11:29"><i><font color="Green" title="E33">A30</font></i></a></i>
;<br/><i><font color="Green" title="E59">A58</font></i>: <font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font> = 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font>
<b>by </b><i><a class="txt" href="#E14:29"><i><font color="Green" title="E36">A35</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">S1</font>,<font color="Maroon" title="c15">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c19">T12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E11:29"><i><font color="Green" title="E33">A30</font></i></a></i>
;<br/><a NAME="E29:29_8"/><i><font color="Green" title="E60">A59</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">a</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E12:29"><i><font color="Green" title="E34">A31</font></i></a></i>;<br/><a NAME="E30:29_8"/><i><font color="Green" title="E61">A60</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/><a NAME="E31:29_8"/><i><font color="Green" title="E62">A61</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E12:29"><i><font color="Green" title="E34">A31</font></i></a></i>;<br/><a NAME="E32:29_8"/><i><font color="Green" title="E63">A62</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c10">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">b</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_8_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c21">T34</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c18">T13</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">T24</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c19">T12</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font></span><br/><a NAME="E1:29_8_3"/><i><font color="Green" title="E64">A63</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c22">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">S4</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c22">c</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E15:29_8"><i><font color="Green" title="E50">A49</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/><a NAME="E2:29_8_3"/><i><font color="Green" title="E65">A64</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c22">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">S2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c22">c</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E13:29_8"><i><font color="Green" title="E48">A47</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/><a NAME="E3:29_8_3"/><i><font color="Green" title="E66">A65</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">S3</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c22">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">S4</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c22">c</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E19:29_8"><i><font color="Green" title="E54">A53</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/><a NAME="E4:29_8_3"/><i><font color="Green" title="E67">A66</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">S1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c22">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">S3</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c22">c</font></span>)</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> 
 <b>by </b><i><a class="txt" href="#E17:29_8"><i><font color="Green" title="E52">A51</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/><a NAME="E5:29_8_3"/><i><font color="Green" title="E68">A67</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c22">c</font>,<font color="Maroon" title="c22">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></i>;<br/><a NAME="E6:29_8_3"/><b>then </b><i><font color="Green" title="E69">A68</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</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> 
 <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="isocat_2.html#T13" target="_self" title="ISOCAT_2:th.13">Th13</a></i>;<br/><a NAME="E7:29_8_3"/><i><font color="Green" title="E70">A69</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</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> 
 <b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/><a NAME="E8:29_8_3"/><i><font color="Green" title="E71">A70</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</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> 
 <b>by </b><i><a class="txt" href="#E6:29_8_3"><i><font color="Green" title="E69">A68</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/><a NAME="E9:29_8_3"/><i><font color="Green" title="E72">A71</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</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> 
 <b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/><a NAME="E10:29_8_3"/><i><font color="Green" title="E73">A72</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</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> 
 <b>by </b><i><a class="txt" href="#E6:29_8_3"><i><font color="Green" title="E69">A68</font></i></a>, <a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">fi</font> = <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c12">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c22">c</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> <b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="txt" href="#E5:29_8_3"><i><font color="Green" title="E68">A67</font></i></a>, <a class="ref" href="cat_2.html#T43" title="CAT_2:th.43">CAT_2:43</a></i>;<br/><a NAME="E12:29_8_3"/>
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#T33" title="CAT_2:th.33">CAT_2:33</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">FF1</font> = <font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c25">FF2</font> = <font color="Maroon" title="c5">F2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C</font> ;<br/><i><font color="Green" title="E74">A73</font></i>: <font color="Maroon" title="c5">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c23">fi</font> = 
<font color="Maroon" title="c25">FF2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c22">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:29_8_3"><i><font color="Green" title="E69">A68</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K3" title="ISOCAT_2:func.3">curry</a> <font color="Maroon" title="c5">F2</font>,<font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c22">c</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_2.html#T3" title="CAT_2:th.3">CAT_2:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="isocat_2.html#K4" title="ISOCAT_2:func.4">?-</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">c</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c20">T24</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font>
<b>by </b><i><a class="txt" href="#E15:29_8"><i><font color="Green" title="E50">A49</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
;<br/><i><font color="Green" title="E75">A74</font></i>: <font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> = 
<font color="Maroon" title="c7">s</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c22">c</font>
<b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">c</font>
<b>by </b><i><a class="ref" href="cat_2.html#T3" title="CAT_2:th.3">CAT_2:3</a></i>
<br/>.= 
<font color="Maroon" title="c19">T12</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font>
<b>by </b><i><a class="txt" href="#E13:29_8"><i><font color="Green" title="E48">A47</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
;<br/><i><font color="Green" title="E76">A75</font></i>: <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c23">fi</font> = 
<font color="Maroon" title="c24">FF1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c22">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:29_8_3"><i><font color="Green" title="E69">A68</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K3" title="ISOCAT_2:func.3">curry</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c22">c</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_2.html#T3" title="CAT_2:th.3">CAT_2:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="isocat_2.html#K4" title="ISOCAT_2:func.4">?-</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">c</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c18">T13</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font>
<b>by </b><i><a class="txt" href="#E17:29_8"><i><font color="Green" title="E52">A51</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
;<br/><i><font color="Green" title="E77">A76</font></i>: <font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> = 
<font color="Maroon" title="c7">s</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c22">c</font>
<b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">c</font>
<b>by </b><i><a class="ref" href="cat_2.html#T3" title="CAT_2:th.3">CAT_2:3</a></i>
<br/>.= 
<font color="Maroon" title="c21">T34</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font>
<b>by </b><i><a class="txt" href="#E19:29_8"><i><font color="Green" title="E54">A53</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c21">T34</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c18">T13</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">T34</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">T13</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E16:29_8"><i><font color="Green" title="E51">A50</font></i></a>, <a class="txt" href="#E18:29_8"><i><font color="Green" title="E53">A52</font></i></a>, <a class="ref" href="nattra_1.html#T27" title="NATTRA_1:th.27">NATTRA_1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c23">fi</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:29_8_3"><i><font color="Green" title="E66">A65</font></i></a>, <a class="txt" href="#E4:29_8_3"><i><font color="Green" title="E67">A66</font></i></a>, <a class="txt" href="#E16:29_8_3"><i><font color="Green" title="E76">A75</font></i></a>, <a class="txt" href="#E17:29_8_3"><i><font color="Green" title="E77">A76</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c23">fi</font></span>)</span>
<b>by </b><i><a class="txt" href="#E9:29_8_3"><i><font color="Green" title="E72">A71</font></i></a>, <a class="txt" href="#E10:29_8_3"><i><font color="Green" title="E73">A72</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c23">fi</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E6:29_8_3"><i><font color="Green" title="E69">A68</font></i></a>, <a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c23">fi</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c22">c</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E7:29_8_3"><i><font color="Green" title="E70">A69</font></i></a>, <a class="txt" href="#E8:29_8_3"><i><font color="Green" title="E71">A70</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">T24</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">T12</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:29_8_3"><i><font color="Green" title="E64">A63</font></i></a>, <a class="txt" href="#E2:29_8_3"><i><font color="Green" title="E65">A64</font></i></a>, <a class="txt" href="#E14:29_8_3"><i><font color="Green" title="E74">A73</font></i></a>, <a class="txt" href="#E15:29_8_3"><i><font color="Green" title="E75">A74</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">T24</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c19">T12</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c22">c</font>
<b>by </b><i><a class="txt" href="#E12:29_8"><i><font color="Green" title="E47">A46</font></i></a>, <a class="txt" href="#E14:29_8"><i><font color="Green" title="E49">A48</font></i></a>, <a class="ref" href="nattra_1.html#T27" title="NATTRA_1:th.27">NATTRA_1:27</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E34:29_8"/><b>then </b><i><font color="Green" title="E64">A77</font></i>: 
<font color="Maroon" title="c21">T34</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c18">T13</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c20">T24</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c19">T12</font>
 <b>by </b><i><a class="txt" href="#E20:29_8"><i><font color="Green" title="E55">A54</font></i></a>, <a class="ref" href="nattra_1.html#T20" title="NATTRA_1:th.20">NATTRA_1:20</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span>
<b>by </b><i><a class="txt" href="#E31:29_8"><i><font color="Green" title="E62">A61</font></i></a>, <a class="txt" href="#E32:29_8"><i><font color="Green" title="E63">A62</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">g</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">S1</font>,<font color="Maroon" title="c17">S4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">T34</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c18">T13</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E25:29_8"><i><font color="Green" title="E56">A55</font></i></a>, <a class="txt" href="#E27:29_8"><i><font color="Green" title="E58">A57</font></i></a>, <a class="ref" href="nattra_1.html#T42" title="NATTRA_1:th.42">NATTRA_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E26:29_8"><i><font color="Green" title="E57">A56</font></i></a>, <a class="txt" href="#E28:29_8"><i><font color="Green" title="E59">A58</font></i></a>, <a class="txt" href="#E34:29_8"><i><font color="Green" title="E64">A77</font></i></a>, <a class="ref" href="nattra_1.html#T42" title="NATTRA_1:th.42">NATTRA_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:29_8"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c10">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E29:29_8"><i><font color="Green" title="E60">A59</font></i></a>, <a class="txt" href="#E30:29_8"><i><font color="Green" title="E61">A60</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>thus </b><a NAME="E18:29"/><i><font color="Green" title="E38">A78</font></i>: 
 <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font>, <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font> st <br/>for <font color="Olive" title="b2">s</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C</font>  st <font color="Maroon" title="c6">t</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">s</font> holds <br/>for <font color="Olive" title="b3">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Olive" title="b1">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b3">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b3">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b3">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_9"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_9"/>
 <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font>
 <b>by </b><i><a class="txt" href="#E13:29"><i><font color="Green" title="E35">A34</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font>, <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></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="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E2:29_9"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font>, <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></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="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E17:29"><i><font color="Green" title="E37">A36</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<a NAME="E19:29"/>
<font color="Maroon" title="c9">G</font> is    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font>, <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_10"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_10"/>
 <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font>
 <b>by </b><i><a class="txt" href="#E18:29"><i><font color="Green" title="E38">A78</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></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="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E2:29_10"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>  holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></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="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E17:29"><i><font color="Green" title="E37">A36</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">G</font> = <font color="Maroon" title="c9">G</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font>, <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c10">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C</font>  st <font color="Maroon" title="c6">t</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">s</font> holds <br/>for <font color="Olive" title="b2">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c10">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b2">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<b>let </b><font color="Maroon" title="c11">s</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">t</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c11">s</font> implies for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c10">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>assume </b><a NAME="E21:29"/><i><font color="Green" title="E39">A79</font></i>: 
<font color="Maroon" title="c6">t</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c11">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c10">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<b>let </b><font color="Maroon" title="c12">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c12">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<b>thus </b><font color="Maroon" title="c10">G</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c12">a</font> = 
<font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
<b>by </b><i><a class="txt" href="#E14:29"><i><font color="Green" title="E36">A35</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c12">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E11:29"><i><font color="Green" title="E33">A30</font></i></a>, <a class="txt" href="#E21:29"><i><font color="Green" title="E39">A79</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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