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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">L1</font> be    <a href="rmod_4.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">L2</font> be    <a href="rmod_4.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>









<b>assume </b><a NAME="E1:46"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rmod_4.html#K2" target="_self">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="rmod_4.html#K6" target="_self">+</a> <font color="Maroon">L2</font></span>)</span>
 ;<br/>

<a NAME="E2:46"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="rmod_4.html#K6" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:46"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E5:46"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="rmod_4.html#K6" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/>
<a NAME="E6:46"/>
( <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="rmod_4.html#K6" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<a NAME="E7:46"/><b>then </b>
( <font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> or <font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:46"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v2</font> where <font color="Olive">v2</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> </span>}</span>  )
 
<b>by </b><i/>;<br/>
<a NAME="E9:46"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">L1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">L2</font> )
 
;<br/>
<b>hence </b><a NAME="E10:46"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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