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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">j1</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b3">j1</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">j1</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">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/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:97"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:97"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:97"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">j1</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><font color="Maroon" title="c10">F9</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/>
<a NAME="E4:97"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
( <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>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="cat_3.html#T79" target="_self" title="CAT_3:th.79">Th85</a></span>;<br/>
<a NAME="E5:97"/>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">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="graph_1.html#K2" title="GRAPH_1:func.2">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="cat_3.html#D17" target="_self" title="CAT_3:def.17">Def19</a></span>;<br/>
<a NAME="E6:97"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A5</font></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> (<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T65" target="_self" title="CAT_3:th.65">Th70</a></span>;<br/>
<a NAME="E7:97"/>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">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="graph_1.html#K2" title="GRAPH_1:func.2">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="cat_3.html#D17" target="_self" title="CAT_3:def.17">Def19</a></span>;<br/>
<a NAME="E8:97"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A6</font></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> (<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T65" target="_self" title="CAT_3:th.65">Th70</a></span>;<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="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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">j1</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">j2</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="cat_3.html#T6" target="_self" title="CAT_3:th.6">Th10</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T6" target="_self" title="CAT_3:th.6">Th10</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:97"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E8:97"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="txt" href="#E6:97"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="cat_3.html#T72" target="_self" title="CAT_3:th.72">Th78</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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