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<b>let </b><font color="Maroon" title="c1">K</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">J</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">L</font>;<br/>





<b>set </b><font color="Maroon" title="c4">J'</font> =  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font>;<br/>
<a NAME="E1:61"/><i><font color="Green" title="E51">A1</font></i>: 
(  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is <a href="mod_4.html#V3" title="MOD_4:attr.3">antimonomorphism</a> iff (  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is <a href="mod_4.html#V1" title="MOD_4:attr.1">antilinear</a> &amp;  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> ) )
 
<b>by </b><i><a class="ref" href="mod_4.html#D8" target="_self" title="MOD_4:def.8">Def8</a></i>;<br/>
<a NAME="E2:61"/>
( <font color="Maroon" title="c3">J</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a> iff  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is <a href="mod_4.html#V1" title="MOD_4:attr.1">antilinear</a> )
 
<b>by </b><i><a class="ref" href="mod_4.html#T35" target="_self" title="MOD_4:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E3:61"/>
( <font color="Maroon" title="c3">J</font> is <a href="mod_4.html#V2" title="MOD_4:attr.2">monomorphism</a> iff  <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is <a href="mod_4.html#V3" title="MOD_4:attr.3">antimonomorphism</a> )
 <b>by </b><i><a class="txt" href="mod_4.html#E1:61"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="mod_4.html#D7" target="_self" title="MOD_4:def.7">Def7</a></i>;<br/>


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