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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:63"/><i><font color="Green">E52</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> iff  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
;<br/>
<a NAME="E2:63"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="ringcat1.html#V1">linear</a> iff  <a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font> is <a href="mod_4.html#V1" target="_self">antilinear</a> )
 
<b>by </b><i><a class="ref" href="mod_4.html#T35" target="_self">Th35</a></i>;<br/>
<b>hence </b><a NAME="E3:63"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="mod_4.html#V4" target="_self">epimorphism</a> iff  <a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font> is <a href="mod_4.html#V5" target="_self">antiepimorphism</a> )
 <b>by </b><i><a class="txt" href="mod_4.html#E1:63"><i><font color="Green">E52</font></i></a>, <a class="ref" href="mod_4.html#D9" target="_self">Def9</a>, <a class="ref" href="mod_4.html#D10" target="_self">Def10</a></i>;<br/>


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