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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">t1</font>, <font color="Olive" title="b4">t2</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>  st  <a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Olive" title="b3">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Olive" title="b4">t2</font> holds <br/><font color="Olive" title="b3">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">t2</font></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">t1</font>, <font color="Olive" title="b2">t2</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>  st  <a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Olive" title="b1">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Olive" title="b2">t2</font> holds <br/><font color="Olive" title="b1">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">t2</font> )</span><br/>







<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E30">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">t1</font>, <font color="Olive" title="b2">t2</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>  st  <a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Olive" title="b1">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Olive" title="b2">t2</font> holds <br/><font color="Olive" title="b1">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">t2</font></span><br/>

<a NAME="E2:33"/><b>then </b><i><font color="Green" title="E31">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">t1</font>, <font color="Maroon" title="c7">t2</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#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c6">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c7">t2</font> implies <font color="Maroon" title="c6">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">t2</font> )</span><br/>

<b>assume </b><a NAME="E3:33"/><i><font color="Green" title="E32">A3</font></i>: 
 <a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c6">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c7">t2</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">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">t2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">ab</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">t1</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t2</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">ab</font></span><br/><b>consider </b><font color="Maroon" title="c9">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c10">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E2:33_1"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c8">ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#T35" title="CAT_2:th.35">CAT_2:35</a></i>;<br/><a NAME="E3:33_1"/>
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="c11">s1</font> = <font color="Maroon" title="c6">t1</font>, <font color="Maroon" title="c12">s2</font> = <font color="Maroon" title="c7">t2</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/><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="c11">s1</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> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c6">t1</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="isocat_2.html#D5" target="_self" title="ISOCAT_2:def.5">Def5</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="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="c12">s2</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>
<b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="#E3:33"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="isocat_2.html#D5" target="_self" title="ISOCAT_2:def.5">Def5</a></i>
;<br/><a NAME="E6:33_1"/><b>then </b><i><font color="Green" title="E34">A5</font></i>: 
<span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c11">s1</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c9">a</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="c12">s2</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>thus </b><font color="Maroon" title="c6">t1</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">ab</font> = 
<font color="Maroon" title="c11">s1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E2:33"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="#E2:33_1"><i><font color="Green" title="E33">A4</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="c12">s2</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E6:33_1"><i><font color="Green" title="E34">A5</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="c12">s2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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="c7">t2</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c8">ab</font>
<b>by </b><i><a class="txt" href="#E2:33"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="#E2:33_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</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>hence </b><a NAME="E5:33"/>
<font color="Maroon" title="c6">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">t2</font>
 <b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="isocat_1.html#T31" title="ISOCAT_1:th.31">ISOCAT_1:31</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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