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<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C</font>;<br/><b>let </b><font color="Maroon" title="c5">F</font> be    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c6">F'</font> be    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c1">I</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c5">F</font>
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c6">F'</font>
 <b>and </b><br/><a NAME="E3:63"/><i><font color="Green" title="E39">A3</font></i>: 
 <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c5">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c6">F'</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E5:63"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font>
 <b>and </b><br/><a NAME="E6:63"/><i><font color="Green" title="E41">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">c</font> holds <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">x</font> ) iff <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:63"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="cat_3.html#E3:63"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E8:63"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>
 <b>and </b><br/><a NAME="E9:63"/><i><font color="Green" title="E43">A7</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> holds <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">x</font> ) iff <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E2:63"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="cat_3.html#E3:63"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></i>;<br/>
<b>thus </b><a NAME="E10:63"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="cat_3.html#E8:63"><i><font color="Green" title="E42">A6</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D17" title="CAT_1:def.17">CAT_1:def 17</a><br/>

<b>reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <b>by </b><i><a class="txt" href="cat_3.html#E8:63"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">g</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D12" title="CAT_1:def.12">CAT_1:def 12</a><br/>

<a NAME="E12:63"/><i><font color="Green" title="E44">A8</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E5:63"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="cat_3.html#E8:63"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>
<b>hence </b><a NAME="E13:63"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c9">g</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c9">g</font> )
 ;<br/>

<a NAME="E14:63"/>
 <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c6">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c6">F'</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">gf</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E16:63"/>
<font color="Maroon" title="c10">gf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c4">d</font>
 <b>and </b><br/><a NAME="E17:63"/><i><font color="Green" title="E45">A9</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c4">d</font> holds <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">x</font> ) iff <font color="Maroon" title="c10">gf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E2:63"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></i>;<br/>
<div><i><font color="Green" title="E46">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c11">k</font> =  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">d</font>;<br/><b>thus </b><a NAME="E1:63_1"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></i>;<br/><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:63_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E3:63_1"/><b>then </b>
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T45" target="_self" title="CAT_3:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E4:63_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</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">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><a NAME="E1:63_2"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E12:63"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/><b>hence </b><a NAME="E2:63_2"/>
<font color="Maroon" title="c9">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c4">d</font>
 ;<br/><b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E3:63_2"/><i><font color="Green" title="E47">A11</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E4:63_2"/><b>then </b>
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T45" target="_self" title="CAT_3:th.45">Th45</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c7">f</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="cat_3.html#E12:63"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="cat_3.html#E8:63"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="cat_3.html#E9:63"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="cat_3.html#E3:63_2"><i><font color="Green" title="E47">A11</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">x</font>
<b>by </b><i><a class="txt" href="cat_3.html#E5:63"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="cat_3.html#E6:63"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="cat_3.html#E3:63_2"><i><font color="Green" title="E47">A11</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><font color="Maroon" title="c9">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c7">f</font> = 
<font color="Maroon" title="c10">gf</font>
<b>by </b><i><a class="txt" href="cat_3.html#E17:63"><i><font color="Green" title="E45">A9</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c9">g</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E12:63"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="cat_3.html#E17:63"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="cat_3.html#E18:63"><i><font color="Green" title="E46">A10</font></i></a></i>
;<br/><br/>

<div><i><font color="Green" title="E47">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c11">k</font> =  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font>;<br/><b>thus </b><a NAME="E1:63_4"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></i>;<br/><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:63_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E3:63_4"/><b>then </b>
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T45" target="_self" title="CAT_3:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E4:63_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:63"/>
 <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c5">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c5">F</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c11">fg</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E24:63"/>
<font color="Maroon" title="c11">fg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c3">c</font>
 <b>and </b><br/><a NAME="E25:63"/><i><font color="Green" title="E48">A13</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c3">c</font> holds <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">x</font> ) iff <font color="Maroon" title="c11">fg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:63"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_5"><b>now </b></a><div class="add"><a NAME="E1:63_5"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E12:63"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/><b>hence </b><a NAME="E2:63_5"/>
<font color="Maroon" title="c7">f</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c3">c</font>
 ;<br/><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E3:63_5"/><i><font color="Green" title="E49">A14</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E4:63_5"/><b>then </b>
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T45" target="_self" title="CAT_3:th.45">Th45</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c9">g</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="txt" href="cat_3.html#E12:63"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="txt" href="cat_3.html#E5:63"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="cat_3.html#E6:63"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="cat_3.html#E3:63_5"><i><font color="Green" title="E49">A14</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="cat_3.html#E8:63"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="cat_3.html#E9:63"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="cat_3.html#E3:63_5"><i><font color="Green" title="E49">A14</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><font color="Maroon" title="c7">f</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c9">g</font> = 
<font color="Maroon" title="c11">fg</font>
<b>by </b><i><a class="txt" href="cat_3.html#E25:63"><i><font color="Green" title="E48">A13</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c9">g</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E12:63"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="cat_3.html#E21:63"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="cat_3.html#E25:63"><i><font color="Green" title="E48">A13</font></i></a></i>
;<br/><br/>


</div>
