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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>;<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>;<br/><b>let </b><font color="Maroon" title="c3">i1</font>, <font color="Maroon" title="c4">i2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:100"/><i><font color="Green" title="E57">A1</font></i>: 
<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>
 <b>and </b><br/><a NAME="E2:100"/><i><font color="Green" title="E58">A2</font></i>: 
(  <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>,<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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <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>,<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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">I</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c6">F</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font>;<br/>
<a NAME="E3:100"/><i><font color="Green" title="E59">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> )
 
<b>by </b><i><a class="ref" href="cat_3.html#T7" target="_self" title="CAT_3:th.7">Th7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>now </b></a><div class="add"><a NAME="E1:100_1"/>
(  <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> &amp;  <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>by </b><i><a class="txt" href="cat_3.html#E1:100"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="cat_3.html#D19" target="_self" title="CAT_3:def.19">Def19</a></i>;<br/><b>hence </b><a NAME="E2:100_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font> is    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c2">c</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="cat_3.html#T70" target="_self" title="CAT_3:th.70">Th70</a></i>;<br/><b>thus </b><a NAME="E3:100_1"/>
<font color="Maroon" title="c2">c</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="cat_3.html#E1:100"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="cat_3.html#T85" target="_self" title="CAT_3:th.85">Th85</a></i>;<br/><b>let </b><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E4:100_1"/>
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 ;<br/><a NAME="E5:100_1"/><b>then </b>
( ( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E6:100_1"/>
 <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x2</font></span>)</span></span>)</span> <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#E2:100"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="cat_3.html#E3:100"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:100"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> is <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a> ) )
 
<b>by </b><i><a class="ref" href="cat_3.html#T79" target="_self" title="CAT_3:th.79">Th79</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:100"/>
( <font color="Maroon" title="c3">i1</font> is <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a> &amp; <font color="Maroon" title="c4">i2</font> is <a href="cat_3.html#V2" title="CAT_3:attr.2">coretraction</a> )
 <b>by </b><i><a class="txt" href="cat_3.html#E3:100"><i><font color="Green" title="E59">A3</font></i></a></i>;<br/>


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