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<b>let </b><font color="Maroon">K</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">L</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">J</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">K</font>,<font color="Maroon">L</font>;<br/>





<b>set </b><font color="Maroon">J'</font> =  <a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font>;<br/>
<a NAME="E1:61"/><i><font color="Green">E26</font></i>: 
(  <a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font> is <a href="mod_4.html#V3" target="_self">antimonomorphism</a> iff (  <a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font> is <a href="mod_4.html#V1" target="_self">antilinear</a> &amp;  <a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font> is <a href="funct_1.html#V2">one-to-one</a> ) )
 
<b>by </b><i><a class="txt" href="mod_4.html#E4"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<a NAME="E2:61"/>
( <font color="Maroon">J</font> is <a href="ringcat1.html#V1">linear</a> iff  <a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font> is <a href="mod_4.html#V1" target="_self">antilinear</a> )
 
<b>by </b><i><a class="ref" href="mod_4.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E3:61"/>
( <font color="Maroon">J</font> is <a href="mod_4.html#V2" target="_self">monomorphism</a> iff  <a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font> is <a href="mod_4.html#V3" target="_self">antimonomorphism</a> )
 <b>by </b><i><a class="txt" href="mod_4.html#E15"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/>


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