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<b>let </b><font color="Maroon" title="c1">K</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">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/>
<b>set </b><font color="Maroon" title="c5">L'</font> =  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">L</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon" title="c3">J</font> is <a href="mod_4.html#V2" title="MOD_4:attr.2">antilinear</a>
 ;<br/><a NAME="E2:66_1"/><b>then </b><i><font color="Green" title="E42">A1</font></i>: 
( <font color="Maroon" title="c3">J</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> &amp; <font color="Maroon" title="c3">J</font> is <a href="mod_4.html#V1" title="MOD_4:attr.1">antimultiplicative</a> &amp; <font color="Maroon" title="c3">J</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a> )
 <b>by </b><i><a class="ref" href="mod_4.html#D7" target="_self" title="MOD_4:def.7">Def7</a></i>;<br/><b>thus </b><a NAME="E3:66_1"/>
 <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:66_1_1"/>
 <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="ringcat1.html#D2" title="RINGCAT1:def.2">RINGCAT1:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E2:66_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>

;<br/>


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

<b>thus </b><a NAME="E2:66_1_1"/>
 <a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font> is <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E2:66_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="mod_4.html#D6" target="_self" title="MOD_4:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E5"><i><font color="Green" title="E4">Lm3</font></i></a></i>
;<br/>


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

<b>thus </b><span class="p1">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="txt" href="mod_4.html#E2:66_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a></i>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">L</font></span>)</span>

; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a><br/>


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

<b>assume </b><a NAME="E2:66"/><i><font color="Green" title="E42">A2</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="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</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="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</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="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ringcat1.html#D2" title="RINGCAT1:def.2">RINGCAT1:def 2</a><br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a>,<a class="ref" href="mod_4.html#D7" target="_self" title="MOD_4:def.7">MOD_4:def 7</a>
<div class="add"><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font>;<br/><b>thus </b><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E2:66"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>

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

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="mod_4.html#D6" target="_self" title="MOD_4:def.6">MOD_4:def 6</a>
<div class="add"><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font>;<br/><b>thus </b><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" title="MOD_4:func.7">opp</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E2:66"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E5"><i><font color="Green" title="E4">Lm3</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<b>thus </b><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">L</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E2:66"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a></i>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">L</font>

; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a><br/>


</div>
