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<div class="add">

<b>let </b><font color="Maroon">RR</font> be   <a href="vectsp_2.html#NM2">domRing</a>;<br/><b>let </b><font color="Maroon">VV</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">RR</font>;<br/><b>let </b><font color="Maroon">LL</font> be    <a href="rmod_4.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">VV</font>;<br/><b>let </b><font color="Maroon">aa</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">RR</font>;<br/>







<b>assume </b><a NAME="E1:53"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font>
 ;<br/>

<b>set </b><font color="Maroon">T</font> = <span class="p1">{<span class="default"> <font color="Olive">uu</font> where <font color="Olive">uu</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font> : <span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">aa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">uu</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">S</font> = <span class="p1">{<span class="default"> <font color="Olive">vv</font> where <font color="Olive">vv</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font> : <font color="Maroon">LL</font> <a href="funct_2.html#K3">.</a> <font color="Olive">vv</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font> </span>}</span> ;<br/>
<a NAME="E2:53"/><i><font color="Green">E49</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">uu</font> where <font color="Olive">uu</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font> : <span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">aa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">uu</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">vv</font> where <font color="Olive">vv</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font> : <font color="Maroon">LL</font> <a href="funct_2.html#K3">.</a> <font color="Olive">vv</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_1"/>
<span class="p1">{<span class="default"> <font color="Olive">uu</font> where <font color="Olive">uu</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font> : <span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">aa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">uu</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">vv</font> where <font color="Olive">vv</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font> : <font color="Maroon">LL</font> <a href="funct_2.html#K3">.</a> <font color="Olive">vv</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>proof </b></a><div class="add">

<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="E1:53_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">uu</font> where <font color="Olive">uu</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font> : <span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">aa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">uu</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">uu</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font><b> such that </b><br/><a NAME="E3:53_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">uu</font>
 <b>and </b><br/><a NAME="E4:53_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">aa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">uu</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font>
 ;<br/>
<a NAME="E5:53_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">aa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">uu</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">uu</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">aa</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:53_1_1"/><b>then </b>
<font color="Maroon">LL</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">uu</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>;<br/>
<b>hence </b><a NAME="E7:53_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">vv</font> where <font color="Olive">vv</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font> : <font color="Maroon">LL</font> <a href="funct_2.html#K3">.</a> <font color="Olive">vv</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font> </span>}</span> 
 <b>by </b><i/>;<br/>


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

<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="E2:53_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">vv</font> where <font color="Olive">vv</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font> : <font color="Maroon">LL</font> <a href="funct_2.html#K3">.</a> <font color="Olive">vv</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">vv</font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font><b> such that </b><br/><a NAME="E4:53_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font>
 <b>and </b><br/><a NAME="E5:53_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">LL</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">vv</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font>
 ;<br/>
<a NAME="E6:53_1"/>
<span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">aa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">vv</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">vv</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">aa</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:53_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">aa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">vv</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font>
 
<b>by </b><i>, , <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:53_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">uu</font> where <font color="Olive">uu</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">VV</font> : <span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">aa</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">uu</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RR</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:53"/>
 <a href="rmod_4.html#K2" target="_self">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">LL</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">aa</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">LL</font>
 <b>by </b><i/>;<br/>


</div>
