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<b>let </b><font color="Maroon" title="c7">h1</font>, <font color="Maroon" title="c8">h2</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_2"/><i><font color="Green" title="E17">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> )
 <b>and </b><br/><a NAME="E2:33_1_2"/><i><font color="Green" title="E18">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c8">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c8">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> )
 ;<br/>

<a NAME="E3:33_1_2"/>
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</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#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c3">b</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; <font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a>  <a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>, <a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="ref" href="cat_4.html#D9" target="_self" title="CAT_4:def.9">Def9</a>, <a class="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">h</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font><b> such that </b><br/><a NAME="E5:33_1_2"/><i><font color="Green" title="E19">A5</font></i>: 
for <font color="Olive" title="b1">h1</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c3">b</font> holds <br/> ( ( <span class="p1">(<span class="default"><a href="cat_4.html#K7" title="CAT_4:func.7">pr1</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K8" title="CAT_4:func.8">pr2</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> ) iff <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="cat_3.html#T60" title="CAT_3:th.60">CAT_3:60</a></i>;<br/>
<a NAME="E6:33_1_2"/>
( <font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> &amp; <font color="Maroon" title="c8">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> )
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:33_1_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="cat_4.html#E2:33_1_2"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="cat_4.html#E5:33_1_2"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:33_1_2"/>
<font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h2</font>
 ;<br/>


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