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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>take </b><font color="Maroon" title="c4">F</font> =  <a href="cat_2.html#K18" title="CAT_2:func.18">pr2</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D4" target="_self" title="ISOCAT_1:def.4">ISOCAT_1:def 4</a><br/>

<b>set </b><font color="Maroon" title="c5">X</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">x11</font>, <font color="Maroon" title="c9">x12</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green" title="E16">A1</font></i>: 
( <font color="Maroon" title="c8">x11</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">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span> &amp; <font color="Maroon" title="c9">x12</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 <font color="Maroon" title="c1">A</font> )
 <b>and </b><br/><a NAME="E4:21_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c6">x1</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"><font color="Maroon" title="c8">x11</font>,<font color="Maroon" title="c9">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>assume </b><a NAME="E5:21_1"/>
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">x21</font>, <font color="Maroon" title="c11">x22</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:21_1"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c10">x21</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">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span> &amp; <font color="Maroon" title="c11">x22</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 <font color="Maroon" title="c1">A</font> )
 <b>and </b><br/><a NAME="E8:21_1"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c7">x2</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"><font color="Maroon" title="c10">x21</font>,<font color="Maroon" title="c11">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">f11</font> = <font color="Maroon" title="c8">x11</font>, <font color="Maroon" title="c13">f21</font> = <font color="Maroon" title="c10">x21</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span> <b>by </b><i><a class="txt" href="isocat_1.html#E3:21_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="isocat_1.html#E7:21_1"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">f12</font> = <font color="Maroon" title="c9">x12</font>, <font color="Maroon" title="c15">f22</font> = <font color="Maroon" title="c11">x22</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="isocat_1.html#E3:21_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="isocat_1.html#E7:21_1"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/><b>assume </b><a NAME="E11:21_1"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font>
 ;<br/><i><font color="Green" title="E21">A6</font></i>: <font color="Maroon" title="c14">f12</font> = 
<font color="Maroon" title="c4">F</font> <a href="cat_2.html#K14" title="CAT_2:func.14">.</a> <font color="Maroon" title="c12">f11</font>,<font color="Maroon" title="c14">f12</font>
<b>by </b><i><a class="ref" href="funct_3.html#D6" title="FUNCT_3:def.6">FUNCT_3:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="cat_2.html#K14" title="CAT_2:func.14">.</a> <font color="Maroon" title="c13">f21</font>,<font color="Maroon" title="c15">f22</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E4:21_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="isocat_1.html#E8:21_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="isocat_1.html#E11:21_1"><i><font color="Green" title="E20">A5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">f22</font>
<b>by </b><i><a class="ref" href="funct_3.html#D6" title="FUNCT_3:def.6">FUNCT_3:def 6</a></i>
;<br/><font color="Maroon" title="c12">f11</font> = 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c13">f21</font>
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/><b>hence </b><a NAME="E14:21_1"/>
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="txt" href="isocat_1.html#E4:21_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="isocat_1.html#E8:21_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="isocat_1.html#E12:21_1"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21"/>
<font color="Maroon" title="c4">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D3" target="_self" title="ISOCAT_1:def.3">ISOCAT_1:def 3</a><br/>

<b>thus </b><a NAME="E3:21"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="isocat_1.html#T2" target="_self" title="ISOCAT_1:th.2">Th2</a></i>;<br/>


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