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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>thus </b><a NAME="E1:9"/>
( <font color="Maroon">c<sub>1</sub></font> is    <a href="modcat_1.html#M5" target="_self">LModMorphism_DOMAIN</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="modcat_1.html#D3" target="_self">Def3</a></i>;<br/>

<b>thus </b><a NAME="E2:9"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) implies <font color="Maroon">c<sub>1</sub></font> is    <a href="modcat_1.html#M5" target="_self">LModMorphism_DOMAIN</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:9_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> is  <a href="mod_2.html#V2">strict</a> <a href="mod_2.html#NM1">LModMorphism</a> of <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="modcat_1.html#M3" target="_self">LModMorphism_DOMAIN</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="modcat_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:9_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="modcat_1.html#M4" target="_self">Element</a> of <font color="Maroon">c<sub>5</sub></font> holds <font color="Olive">b<sub>1</sub></font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="modcat_1.html#E1:9_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:9_1"/>
<font color="Maroon">c<sub>1</sub></font> is    <a href="modcat_1.html#M5" target="_self">LModMorphism_DOMAIN</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="modcat_1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>


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