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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="modcat_1.html#M1" target="_self">LeftMod_DOMAIN</a> of <font color="Maroon">R</font>;<br/>



<b>set </b><font color="Maroon">X</font> =  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">V</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">V</font>,   <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">V</font>] <b>means </b> <a href="modcat_1.html#K5" target="_self">dom'</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="modcat_1.html#K6" target="_self">cod'</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>let </b><font color="Maroon">g</font> be    <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">V</font>, <font color="Maroon">f</font> be    <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font>,<font color="Maroon">f</font>]
 ;<br/>

<b>consider </b><font color="Maroon">G2</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="modcat_1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">G3</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="modcat_1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:25"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">g</font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">G2</font>,<font color="Maroon">G3</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:25"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="modcat_1.html#K5" target="_self">dom'</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="mod_2.html#D11">MOD_2:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">G1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="modcat_1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">G2'</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="modcat_1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:25"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2'</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:25"/>
<font color="Maroon">G2'</font> <a href="hidden.html#R1">=</a>  <a href="modcat_1.html#K6" target="_self">cod'</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="mod_2.html#D11">MOD_2:def 11</a></i>;<br/>
<b>hence </b><a NAME="E8:25"/>
 ex <font color="Olive">G1</font>, <font color="Olive">G2</font>, <font color="Olive">G3</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="modcat_1.html#M2" target="_self">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">g</font> is    <a href="mod_2.html#M1">Morphism</a> of <font color="Olive">G2</font>,<font color="Olive">G3</font> &amp; <font color="Maroon">f</font> is    <a href="mod_2.html#M1">Morphism</a> of <font color="Olive">G1</font>,<font color="Olive">G2</font> )
 <b>by </b><i>, , , </i>;<br/>


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