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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="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>thus </b><a NAME="E1:11"/>
 <a href="rlvect_1.html#G1">LoopStr</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>1</sub></font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font> #)
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V4">add-associative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V5">right_zeroed</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V6">right_complementable</a> )
 ;<br/><a NAME="E2:11_1"/><i><font color="Green">E8</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <span class="p2">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:11_1"/><i><font color="Green">E9</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:11_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <span class="p2">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:11_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E4:11_1_1"/><i><font color="Green">E10</font></i>: 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="mod_4.html#E1:11_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="mod_4.html#E7"><i><font color="Green">Lemma6</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <span class="p2">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D25">VECTSP_1:def 25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="mod_4.html#E4:11_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="vectsp_1.html#D25">VECTSP_1:def 25</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:11_1"/>
 <a href="vectsp_1.html#K7">comp</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="mod_4.html#E2:11_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="mod_4.html#E3:11_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E3:11"/>
( <font color="Maroon">c<sub>2</sub></font> is   <a href="vectsp_1.html#NM3">Scalar</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> iff <font color="Maroon">c<sub>2</sub></font> is   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


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