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

<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#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> implies ( <font color="Maroon" title="c2">L</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a> ) )</span><br/>

<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">L</font> is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a> )</span><br/>

<a NAME="E2:28"/><b>then </b>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> holds  <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D18" target="_self" title="VECTSP_1:def.18">Def18</a></i>;<br/>
<b>hence </b><a NAME="E3:28"/>
<font color="Maroon" title="c2">L</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D11" target="_self" title="VECTSP_1:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> is <a href="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a></span><br/>

<a NAME="E4:28"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:28"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="vectsp_1.html#D18" target="_self" title="VECTSP_1:def.18">Def18</a></i>;<br/>
<b>hence </b><a NAME="E5:28"/>
<font color="Maroon" title="c2">L</font> is <a href="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D12" target="_self" title="VECTSP_1:def.12">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
