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<span class="kw">thus </span><a NAME="E1:58_1_1"/>
 <a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="isocat_1.html#D3" title="ISOCAT_1:def.3">ISOCAT_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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"><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></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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> or  not <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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> or  not <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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#K15" title="ISOCAT_2:func.15">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> or <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1_1"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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> or  not <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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#K15" title="ISOCAT_2:func.15">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> or <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:58_1_1_1"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:58_1_1_1"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E6:58_1_1_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">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>
 <span class="kw">and </span><br/><a NAME="E7:58_1_1_1"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="isocat_2.html#T8" target="_self" title="ISOCAT_2:th.8">Th9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:58_1_1_1"/>
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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#K15" title="ISOCAT_2:func.15">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> or <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E11:58_1_1_1"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E12:58_1_1_1"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E13:58_1_1_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">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>
 <span class="kw">and </span><br/><a NAME="E14:58_1_1_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="isocat_2.html#T8" target="_self" title="ISOCAT_2:th.8">Th9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E15:58_1_1_1"/>
<span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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#K15" title="ISOCAT_2:func.15">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font></span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E53">A5</font></span>: <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#K9" title="ISOCAT_2:func.9">Pr1</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">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#K11" title="ISOCAT_2:func.11">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#K10" title="ISOCAT_2:func.10">Pr2</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">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#K12" title="ISOCAT_2:func.12">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#K15" title="ISOCAT_2:func.15">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E7:58_1_1_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E14:58_1_1_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="isocat_2.html#D13" target="_self" title="ISOCAT_2:def.13">Def13</a></span>
<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#K9" title="ISOCAT_2:func.9">Pr1</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">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#K11" title="ISOCAT_2:func.11">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#K10" title="ISOCAT_2:func.10">Pr2</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p4">(<span class="default"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">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#K12" title="ISOCAT_2:func.12">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="isocat_2.html#D13" target="_self" title="ISOCAT_2:def.13">Def13</a></span>
;<br/>
<a NAME="E17:58_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<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#K9" title="ISOCAT_2:func.9">Pr1</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">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#K11" title="ISOCAT_2:func.11">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#K9" title="ISOCAT_2:func.9">Pr1</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">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#K11" title="ISOCAT_2:func.11">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="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E18:58_1_1_1"/><span class="lab"><font color="Green" title="E55">A7</font></span>: 
<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#K10" title="ISOCAT_2:func.10">Pr2</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">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#K12" title="ISOCAT_2:func.12">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#K10" title="ISOCAT_2:func.10">Pr2</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">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#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_1_1_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E19:58_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A8</font></span>: 
 <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">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#K12" title="ISOCAT_2:func.12">Pr2</a> <font color="Maroon" title="c13">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E20:58_1_1_1"/><span class="lab"><font color="Green" title="E57">A9</font></span>: 
<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#K10" title="ISOCAT_2:func.10">Pr2</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">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#K10" title="ISOCAT_2:func.10">Pr2</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">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="kw">by</span> <span class="lab"><a class="txt" href="#E18:58_1_1_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E21:58_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A10</font></span>: 
 <a href="isocat_2.html#K10" title="ISOCAT_2:func.10">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#K10" title="ISOCAT_2:func.10">Pr2</a> <font color="Maroon" title="c11">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E22:58_1_1_1"/><span class="lab"><font color="Green" title="E59">A11</font></span>: 
<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#K9" title="ISOCAT_2:func.9">Pr1</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">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#K9" title="ISOCAT_2:func.9">Pr1</a> <font color="Maroon" title="c11">G1</font></span>)</span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">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="kw">by</span> <span class="lab"><a class="txt" href="#E17:58_1_1_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E23:58_1_1_1"/><span class="kw">then </span>
 <a href="isocat_2.html#K9" title="ISOCAT_2:func.9">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#K9" title="ISOCAT_2:func.9">Pr1</a> <font color="Maroon" title="c11">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E24:58_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A12</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:58_1_1_1"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="ref" href="isocat_2.html#T32" target="_self" title="ISOCAT_2:th.32">Th37</a></span>;<br/>
<a NAME="E25:58_1_1_1"/>
 <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">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#K11" title="ISOCAT_2:func.11">Pr1</a> <font color="Maroon" title="c13">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:58_1_1_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E26:58_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A13</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E11:58_1_1_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E19:58_1_1_1"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="ref" href="isocat_2.html#T33" target="_self" title="ISOCAT_2:th.33">Th39</a></span>;<br/>
<a NAME="E27:58_1_1_1"/><span class="lab"><font color="Green" title="E62">A14</font></span>: 
 <a href="isocat_2.html#K10" title="ISOCAT_2:func.10">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#K10" title="ISOCAT_2:func.10">Pr2</a> <font color="Maroon" title="c12">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:58_1_1_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E28:58_1_1_1"/>
 <a href="isocat_2.html#K9" title="ISOCAT_2:func.9">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#K9" title="ISOCAT_2:func.9">Pr1</a> <font color="Maroon" title="c12">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:58_1_1_1"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:58_1_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E14:58_1_1_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E27:58_1_1_1"><span class="lab"><font color="Green" title="E62">A14</font></span></a>, <a class="txt" href="#E26:58_1_1_1"><span class="lab"><font color="Green" title="E61">A13</font></span></a>, <a class="txt" href="#E24:58_1_1_1"><span class="lab"><font color="Green" title="E60">A12</font></span></a>, <a class="ref" href="isocat_2.html#T32" target="_self" title="ISOCAT_2:th.32">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:58_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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="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"><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>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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"><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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">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"><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> or <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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> )</span><br/>

<span class="kw">assume </span><a NAME="E3:58_1_1"/>
<font color="Maroon" title="c4">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"><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:58_1_1"/><span class="lab"><font color="Green" title="E49">A15</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T27" title="CAT_2:th.27">CAT_2:27</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">G1</font>, <font color="Maroon" title="c8">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="c9">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">G1</font>,<font color="Maroon" title="c8">G2</font><span class="kw"> such that </span><br/><a NAME="E7:58_1_1"/><span class="lab"><font color="Green" title="E50">A16</font></span>: 
<font color="Maroon" title="c7">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">G2</font>
 <span class="kw">and </span><br/><a NAME="E8:58_1_1"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G1</font>
 <span class="kw">and </span><br/><a NAME="E9:58_1_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G2</font>
 <span class="kw">and </span><br/><a NAME="E10:58_1_1"/><span class="lab"><font color="Green" title="E51">A17</font></span>: 
<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="c7">G1</font>,<font color="Maroon" title="c8">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="isocat_2.html#T8" target="_self" title="ISOCAT_2:th.8">Th9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">F1</font>, <font color="Maroon" title="c11">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="c12">s</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font><span class="kw"> such that </span><br/><a NAME="E12:58_1_1"/><span class="lab"><font color="Green" title="E52">A18</font></span>: 
<font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">F2</font>
 <span class="kw">and </span><br/><a NAME="E13:58_1_1"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F1</font>
 <span class="kw">and </span><br/><a NAME="E14:58_1_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F2</font>
 <span class="kw">and </span><br/><a NAME="E15:58_1_1"/><span class="lab"><font color="Green" title="E53">A19</font></span>: 
<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="c10">F1</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="isocat_2.html#T8" target="_self" title="ISOCAT_2:th.8">Th9</a></span>;<br/>
<span class="kw">set </span><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#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c7">G1</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&gt;</a></span>,<span class="p3"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c11">F2</font>,<font color="Maroon" title="c8">G2</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c9">t</font></span><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">:&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E16:58_1_1"/><span class="lab"><font color="Green" title="E54">A20</font></span>: 
<span class="p1"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c7">G1</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&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#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c11">F2</font>,<font color="Maroon" title="c8">G2</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:58_1_1"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="ref" href="isocat_2.html#T38" target="_self" title="ISOCAT_2:th.38">Th46</a></span>;<br/>
<a NAME="E17:58_1_1"/><span class="kw">then </span>
<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#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c7">G1</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&gt;</a></span>,<span class="p3"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c11">F2</font>,<font color="Maroon" title="c8">G2</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c9">t</font></span><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">:&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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T32" title="NATTRA_1:th.32">NATTRA_1:32</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c7">G1</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&gt;</a></span>,<span class="p3"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c11">F2</font>,<font color="Maroon" title="c8">G2</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c9">t</font></span><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">:&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> <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D17" title="NATTRA_1:def.17">NATTRA_1:def 17</a></span>;<br/>
<a NAME="E19:58_1_1"/><span class="lab"><font color="Green" title="E55">A21</font></span>: 
(  <a href="isocat_2.html#K9" title="ISOCAT_2:func.9">Pr1</a> <span class="p1"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c7">G1</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">F1</font> &amp;  <a href="isocat_2.html#K9" title="ISOCAT_2:func.9">Pr1</a> <span class="p1"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c11">F2</font>,<font color="Maroon" title="c8">G2</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">F2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="isocat_2.html#T31" target="_self" title="ISOCAT_2:th.31">Th36</a></span>;<br/>
<a NAME="E20:58_1_1"/><span class="lab"><font color="Green" title="E56">A22</font></span>: 
(  <a href="isocat_2.html#K10" title="ISOCAT_2:func.10">Pr2</a> <span class="p1"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c7">G1</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">G1</font> &amp;  <a href="isocat_2.html#K10" title="ISOCAT_2:func.10">Pr2</a> <span class="p1"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">&lt;:</a><span class="default"><font color="Maroon" title="c11">F2</font>,<font color="Maroon" title="c8">G2</font></span><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="isocat_2.html#T31" target="_self" title="ISOCAT_2:th.31">Th36</a></span>;<br/>
<a NAME="E21:58_1_1"/>
(  <a href="isocat_2.html#K11" title="ISOCAT_2:func.11">Pr1</a> <span class="p1"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c9">t</font></span><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">s</font> &amp;  <a href="isocat_2.html#K12" title="ISOCAT_2:func.12">Pr2</a> <span class="p1"><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c9">t</font></span><a href="isocat_2.html#K14" title="ISOCAT_2:func.14">:&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">t</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:58_1_1"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="ref" href="isocat_2.html#T39" target="_self" title="ISOCAT_2:th.39">Th47</a></span>;<br/>
<a NAME="E22:58_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="txt" href="#E15:58_1_1"><span class="lab"><font color="Green" title="E53">A19</font></span></a>, <a class="txt" href="#E10:58_1_1"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="txt" href="#E16:58_1_1"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="txt" href="#E19:58_1_1"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="txt" href="#E20:58_1_1"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="ref" href="isocat_2.html#D13" target="_self" title="ISOCAT_2:def.13">Def13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:58_1_1"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T112" title="FUNCT_2:th.112">FUNCT_2:112</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
