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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_4.html#NM2" title="CAT_4:NM.2">Cocartesian_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">a</font>, <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">b</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="b4">f</font>, <font color="Olive" title="b5">k</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">c</font><br/> for <font color="Olive" title="b6">g</font>, <font color="Olive" title="b7">h</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">c</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">c</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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">c</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>  &amp; <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Olive" title="b4">f</font>,<font color="Olive" title="b6">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Olive" title="b5">k</font>,<font color="Olive" title="b7">h</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> holds <br/>( <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">k</font> &amp; <font color="Olive" title="b6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">h</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">b</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">f</font>, <font color="Olive" title="b2">k</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">c</font><br/> for <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">c</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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</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>  &amp; <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">h</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> holds <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">h</font> )</span><br/><b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">k</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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">g</font>, <font color="Olive" title="b2">h</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">c</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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</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>  &amp; <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Olive" title="b1">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c6">k</font>,<font color="Olive" title="b2">h</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> holds <br/>( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> )</span><br/><b>let </b><font color="Maroon" title="c7">g</font>, <font color="Maroon" title="c8">h</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">c</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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</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>  &amp; <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c8">h</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> implies ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> ) )</span><br/>











<b>assume </b><a NAME="E1:97"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">c</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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c8">h</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> or ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> ) )</span><br/>

<a NAME="E2:97"/><b>then </b>
( <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp; <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c8">h</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> &amp; <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> &amp; <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c8">h</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> )
 
<b>by </b><i><a class="ref" href="cat_4.html#D29" target="_self" title="CAT_4:def.29">Def29</a></i>;<br/>
<b>hence </b><a NAME="E3:97"/>
(  not <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c8">h</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> or ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> ) )
 ; <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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