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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>, <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="c1">C</font><br/> for <font color="Olive" title="b3">i1</font>, <font color="Olive" title="b4">i2</font>, <font color="Olive" title="b5">j1</font>, <font color="Olive" title="b6">j2</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="b3">i1</font>,<font color="Olive" title="b4">i2</font> &amp; <font color="Olive" title="b2">d</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b5">j1</font>,<font color="Olive" title="b6">j2</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b5">j1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b6">j2</font> holds <br/><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a> </span><br/><b>let </b><font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c3">d</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">j1</font>, <font color="Olive" title="b4">j2</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="Maroon" title="c3">d</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b3">j1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b4">j2</font> holds <br/><font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a> </span><br/><b>let </b><font color="Maroon" title="c4">i1</font>, <font color="Maroon" title="c5">i2</font>, <font color="Maroon" title="c6">j1</font>, <font color="Maroon" title="c7">j2</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="c4">i1</font>,<font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c3">d</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c7">j2</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">j1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">j2</font> implies <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a>  )</span><br/>













<b>assume </b><b>that </b><br/><a NAME="E1:98"/><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="c4">i1</font>,<font color="Maroon" title="c5">i2</font>
 <b>and </b><br/><a NAME="E2:98"/><i><font color="Green" title="E58">A2</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c7">j2</font>
 <b>and </b><br/><a NAME="E3:98"/><i><font color="Green" title="E59">A3</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">j1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">j2</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>,<font color="Maroon" title="c3">d</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a> </span><br/>

<b>set </b><font color="Maroon" title="c8">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="c9">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="c4">i1</font>,<font color="Maroon" title="c5">i2</font>;<br/>
<b>set </b><font color="Maroon" title="c10">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="c6">j1</font>,<font color="Maroon" title="c7">j2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>now </b></a><div class="add"><a NAME="E1:98_1"/>
(  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">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="c5">i2</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="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:98"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="cat_3.html#E2:98"><i><font color="Green" title="E58">A2</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="E2:98_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="c4">i1</font>,<font color="Maroon" title="c5">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> &amp; <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="c6">j1</font>,<font color="Maroon" title="c7">j2</font> is    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c3">d</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>; <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#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="c4">i1</font>,<font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c3">d</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="c6">j1</font>,<font color="Maroon" title="c7">j2</font> &amp;  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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="c4">i1</font>,<font color="Maroon" title="c5">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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="c6">j1</font>,<font color="Maroon" title="c7">j2</font></span>)</span> )</span><br/><b>thus </b><a NAME="E3:98_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="c4">i1</font>,<font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c3">d</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="c6">j1</font>,<font color="Maroon" title="c7">j2</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:98"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="cat_3.html#E2:98"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="cat_3.html#T85" target="_self" title="CAT_3:th.85">Th85</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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="c4">i1</font>,<font color="Maroon" title="c5">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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="c6">j1</font>,<font color="Maroon" title="c7">j2</font></span>)</span></span><br/><b>thus </b> <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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="c4">i1</font>,<font color="Maroon" title="c5">i2</font></span>)</span> = 
<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> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">j1</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="c7">j2</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E3:98"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="cat_3.html#T10" target="_self" title="CAT_3:th.10">Th10</a></i>
<br/>.= 
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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="c6">j1</font>,<font color="Maroon" title="c7">j2</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_3.html#T10" target="_self" title="CAT_3:th.10">Th10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:98"/>
<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="cat_3.html#T78" target="_self" title="CAT_3:th.78">Th78</a></i>; <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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