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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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><br/> for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Olive" title="b1">c</font>,<font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b4">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">i1</font> &amp; <font color="Olive" title="b4">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> holds <br/><font color="Olive" title="b4">h</font> <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="b1">c</font></span><br/><b>let </b><font color="Maroon" title="c2">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Maroon" title="c2">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b3">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i1</font> &amp; <font color="Olive" title="b3">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b2">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">i2</font> holds <br/><font color="Olive" title="b3">h</font> <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="c2">c</font></span><br/><b>let </b><font color="Maroon" title="c3">i1</font>, <font color="Maroon" title="c4">i2</font>, <font color="Maroon" title="c5">h</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c5">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font> &amp; <font color="Maroon" title="c5">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> implies <font color="Maroon" title="c5">h</font> <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="c2">c</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:101"/><i><font color="Green" title="E58">A1</font></i>: 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font>
 <b>and </b><br/><a NAME="E2:101"/><i><font color="Green" title="E59">A2</font></i>: 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font>
 <b>and </b><br/><a NAME="E3:101"/><i><font color="Green" title="E60">A3</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c3">i1</font></span>)</span>,<font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">i2</font></span>)</span>,<font color="Olive" title="b1">d</font> holds <br/> ex <font color="Olive" title="b4">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">c</font>,<font color="Olive" title="b1">d</font> &amp; ( for <font color="Olive" title="b5">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">c</font>,<font color="Olive" title="b1">d</font> holds <br/>( ( <font color="Olive" title="b5">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b5">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) iff <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">k</font> ) ) )
 <b>and </b><br/><a NAME="E4:101"/><i><font color="Green" title="E61">A4</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c2">c</font>
 <b>and </b><br/><a NAME="E5:101"/><i><font color="Green" title="E62">A5</font></i>: 
( <font color="Maroon" title="c5">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font> &amp; <font color="Maroon" title="c5">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D19" target="_self" title="CAT_3:def.19">CAT_3:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">h</font> <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="c2">c</font></span><br/>

<a NAME="E6:101"/>
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c3">i1</font></span>)</span>,<font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">i2</font></span>)</span>,<font color="Maroon" title="c2">c</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:101"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="cat_3.html#E2:101"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">i</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E8:101"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c2">c</font>
 <b>and </b><br/><a NAME="E9:101"/><i><font color="Green" title="E63">A6</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="c1">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#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c2">c</font> holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font> &amp; <font color="Olive" title="b1">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> ) iff <font color="Maroon" title="c6">i</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#E3:101"><i><font color="Green" title="E60">A3</font></i></a></i>;<br/>
<a NAME="E10:101"/>
(  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c2">c</font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:101"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="cat_3.html#E2:101"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a>, <a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></i>;<br/>
<b>hence </b> <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">c</font> = 
<font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="cat_3.html#E9:101"><i><font color="Green" title="E63">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">h</font>
<b>by </b><i><a class="txt" href="cat_3.html#E4:101"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="cat_3.html#E5:101"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="cat_3.html#E9:101"><i><font color="Green" title="E63">A6</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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