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<b>let </b><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c3">T</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font>;<br/>





<b>assume </b><a NAME="E1:85"/><i><font color="Green" title="E45">A1</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b2">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> st <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b2">d</font>
 ;<br/>

<a NAME="E2:85"/><i><font color="Green" title="E46">A2</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">D</font> st <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>consider </b><font color="Maroon" title="c5">d</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E2:85_1"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:85"><i><font color="Green" title="E45">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">d</font>
;<br/>

<b>thus </b><a NAME="E3:85_1"/>
<font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E2:85_1"><i><font color="Green" title="E47">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c4">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c5">d</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font>;<br/>



<b>thus </b><a NAME="E3:85"/>
( <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">d</font> implies <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="txt" href="cat_1.html#E2:85"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="cat_1.html#D19" target="_self" title="CAT_1:def.19">Def19</a></i>;<br/>


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