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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>;<br/>





<b>thus </b><a NAME="E1:60"/>
 <a href="isocat_2.html#K17" title="ISOCAT_2:func.17">distribute</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D3" title="ISOCAT_1:def.3">ISOCAT_1:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K17" title="ISOCAT_2:func.17">distribute</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">f1</font> = <font color="Maroon" title="c4">x1</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</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="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c7">F1</font>, <font color="Maroon" title="c8">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>, <font color="Maroon" title="c9">s</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c8">F2</font><b> such that </b><br/><a NAME="E4:60_1"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c7">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">F2</font>
 <b>and </b><br/><a NAME="E5:60_1"/><i><font color="Green" title="E53">A2</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F2</font> &amp; <font color="Maroon" title="c6">f1</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="c7">F1</font>,<font color="Maroon" title="c8">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self" title="ISOCAT_2:th.9">Th9</a></i>;<br/>
<b>assume </b><a NAME="E6:60_1"/>
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K17" title="ISOCAT_2:func.17">distribute</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">f2</font> = <font color="Maroon" title="c5">x2</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</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="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c11">G1</font>, <font color="Maroon" title="c12">G2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>, <font color="Maroon" title="c13">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c11">G1</font>,<font color="Maroon" title="c12">G2</font><b> such that </b><br/><a NAME="E9:60_1"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c11">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c12">G2</font>
 <b>and </b><br/><a NAME="E10:60_1"/><i><font color="Green" title="E55">A4</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">G1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G2</font> &amp; <font color="Maroon" title="c10">f2</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="c11">G1</font>,<font color="Maroon" title="c12">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self" title="ISOCAT_2:th.9">Th9</a></i>;<br/>
<b>assume </b><a NAME="E11:60_1"/>
<span class="p1">(<span class="default"><a href="isocat_2.html#K17" title="ISOCAT_2:func.17">distribute</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K17" title="ISOCAT_2:func.17">distribute</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font>
 ;<br/>

<b>then </b><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"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c8">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c9">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c8">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c9">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K17" title="ISOCAT_2:func.17">distribute</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c11">G1</font>,<font color="Maroon" title="c12">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E4:60_1"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="isocat_2.html#E5:60_1"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="isocat_2.html#E10:60_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="isocat_2.html#D13" target="_self" title="ISOCAT_2:def.13">Def13</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"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c12">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c12">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E9:60_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="isocat_2.html#D13" target="_self" title="ISOCAT_2:def.13">Def13</a></i>
;<br/>
<a NAME="E13:60_1"/><b>then </b>
( <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"><span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c8">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c9">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c12">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <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"><span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c8">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c9">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c12">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E14:60_1"/><b>then </b><i><font color="Green" title="E56">A5</font></i>: 
(  <a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c9">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <font color="Maroon" title="c13">t</font> &amp;  <a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c9">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <font color="Maroon" title="c13">t</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c8">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c12">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c8">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c12">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E15:60_1"/><b>then </b>
(  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c7">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c11">G1</font> &amp;  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c7">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c11">G1</font> &amp;  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c8">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c12">G2</font> &amp;  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c8">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c12">G2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E16:60_1"/><b>then </b>
( <font color="Maroon" title="c9">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">t</font> &amp; <font color="Maroon" title="c7">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">G1</font> &amp; <font color="Maroon" title="c8">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">G2</font> )
 
<b>by </b><i><a class="txt" href="isocat_2.html#E4:60_1"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="isocat_2.html#E9:60_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="isocat_2.html#E14:60_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="isocat_2.html#T37" target="_self" title="ISOCAT_2:th.37">Th37</a>, <a class="ref" href="isocat_2.html#T39" target="_self" title="ISOCAT_2:th.39">Th39</a></i>;<br/>
<b>hence </b><a NAME="E17:60_1"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <b>by </b><i><a class="txt" href="isocat_2.html#E5:60_1"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="isocat_2.html#E10:60_1"><i><font color="Green" title="E55">A4</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:60"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K17" title="ISOCAT_2:func.17">distribute</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c4">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:60"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">o1</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</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="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>, <font color="Maroon" title="c6">o2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</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="c1">A</font>,<font color="Maroon" title="c3">C</font></span>)</span><b> such that </b><br/><a NAME="E5:60"/><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#T37" title="CAT_2:th.37">CAT_2:37</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">F1</font>, <font color="Maroon" title="c8">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c9">s</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c8">F2</font><b> such that </b><br/><a NAME="E7:60"/><i><font color="Green" title="E53">A7</font></i>: 
<font color="Maroon" title="c7">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">F2</font>
 <b>and </b><br/><a NAME="E8:60"/><i><font color="Green" title="E54">A8</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F2</font> &amp; <font color="Maroon" title="c5">o1</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="c7">F1</font>,<font color="Maroon" title="c8">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self" title="ISOCAT_2:th.9">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">G1</font>, <font color="Maroon" title="c11">G2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c12">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">G1</font>,<font color="Maroon" title="c11">G2</font><b> such that </b><br/><a NAME="E10:60"/><i><font color="Green" title="E55">A9</font></i>: 
<font color="Maroon" title="c10">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">G2</font>
 <b>and </b><br/><a NAME="E11:60"/><i><font color="Green" title="E56">A10</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">G1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">G2</font> &amp; <font color="Maroon" title="c6">o2</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">G1</font>,<font color="Maroon" title="c11">G2</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="ref" href="isocat_2.html#T9" target="_self" title="ISOCAT_2:th.9">Th9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">f</font> = <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"><span class="p3"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c10">G1</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span>,<span class="p3"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c8">F2</font>,<font color="Maroon" title="c11">G2</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">&lt;:</a><span class="default"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c12">t</font></span><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">:&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E12:60"/><i><font color="Green" title="E57">A11</font></i>: 
<span class="p1"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c10">G1</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <span class="p1"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c8">F2</font>,<font color="Maroon" title="c11">G2</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E7:60"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="isocat_2.html#E10:60"><i><font color="Green" title="E55">A9</font></i></a>, <a class="ref" href="isocat_2.html#T46" target="_self" title="ISOCAT_2:th.46">Th46</a></i>;<br/>
<a NAME="E13:60"/><b>then </b>
<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"><span class="p3"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c10">G1</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span>,<span class="p3"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c8">F2</font>,<font color="Maroon" title="c11">G2</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">&lt;:</a><span class="default"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c12">t</font></span><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">:&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">A</font>,<span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#T35" title="NATTRA_1:th.35">NATTRA_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">f</font> = <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"><span class="p3"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c10">G1</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span>,<span class="p3"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c8">F2</font>,<font color="Maroon" title="c11">G2</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">&lt;:</a><span class="default"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c12">t</font></span><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">:&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</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="c1">A</font>,<span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span></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>;<br/>
<a NAME="E15:60"/>
(  <a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <span class="p1"><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">&lt;:</a><span class="default"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c12">t</font></span><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">s</font> &amp;  <a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <span class="p1"><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">&lt;:</a><span class="default"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c12">t</font></span><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">t</font> &amp;  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <span class="p1"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c10">G1</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">F1</font> &amp;  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <span class="p1"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c8">F2</font>,<font color="Maroon" title="c11">G2</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">F2</font> &amp;  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <span class="p1"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c10">G1</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">G1</font> &amp;  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <span class="p1"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c8">F2</font>,<font color="Maroon" title="c11">G2</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">G2</font> )
 
<b>by </b><i><a class="txt" href="isocat_2.html#E7:60"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="isocat_2.html#E10:60"><i><font color="Green" title="E55">A9</font></i></a>, <a class="ref" href="isocat_2.html#T36" target="_self" title="ISOCAT_2:th.36">Th36</a>, <a class="ref" href="isocat_2.html#T47" target="_self" title="ISOCAT_2:th.47">Th47</a></i>;<br/>
<a NAME="E16:60"/><b>then </b>
<span class="p1">(<span class="default"><a href="isocat_2.html#K17" title="ISOCAT_2:func.17">distribute</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E5:60"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="isocat_2.html#E8:60"><i><font color="Green" title="E54">A8</font></i></a>, <a class="txt" href="isocat_2.html#E11:60"><i><font color="Green" title="E56">A10</font></i></a>, <a class="txt" href="isocat_2.html#E12:60"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="isocat_2.html#D13" target="_self" title="ISOCAT_2:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E17:60"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K17" title="ISOCAT_2:func.17">distribute</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T4" target="_self" title="ISOCAT_2:th.4">Th4</a></i>;<br/>


</div>
