<?xml version="1.0"?>
<div class="add">

<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/>
<b>set </b><font color="Maroon">L'</font> =  <a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">L</font>;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">J</font> is <a href="ringcat1.html#V1">linear</a>
 ;<br/><a NAME="E2:59_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font> holds  <span class="p1">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font>, <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font>;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_4.html#T7" target="_self">Th7</a>, <a class="ref" href="ringcat1.html#D2">RINGCAT1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>

;<br/>


</div><b>end;</b></div><a NAME="E3:59_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font> holds  <span class="p1">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font>, <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font>;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_4.html#T7" target="_self">Th7</a>, <a class="ref" href="ringcat1.html#D2">RINGCAT1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> = 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="mod_4.html#T7" target="_self">Th7</a>, <a class="ref" href="ringcat1.html#D2">RINGCAT1:def 2</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E5:59_1"/>
 <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="txt" href="mod_4.html#E15"><i><font color="Green">Lemma51</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_2"/><i><font color="Green">E82</font></i>: 
 <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>
 ;<br/><a NAME="E2:59_2"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font> holds  <font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font>, <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font>;<br/>

<b>thus </b><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_4.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>

;<br/>


</div><b>end;</b></div><a NAME="E3:59_2"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font> holds  <font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font>, <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font>;<br/>

<b>thus </b><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K7" target="_self">opp</a> <font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_4.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> = 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_4.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E5:59_2"/>
<font color="Maroon">J</font> is <a href="ringcat1.html#V1">linear</a>
 <b>by </b><i><a class="txt" href="mod_4.html#E15"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="ringcat1.html#D2">RINGCAT1:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:59"/>
( <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#T6" target="_self">Th6</a></i>;<br/>


</div>
