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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>thus </b><a NAME="E1:14"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlsub_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlsub_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E4:14_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T79">RLSUB_1:79</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:14_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_4.html#E3:14_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="rlvect_4.html#T11" target="_self">Th11</a></i>;<br/>
<b>thus </b><a NAME="E7:14_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_4.html#E4:14_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rlvect_4.html#E6:14_1"><i><font color="Green">E8</font></i></a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E2:14"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:14"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_4.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlsub_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="rlvect_4.html#E2:14"><i><font color="Green">E6</font></i></a>, <a class="ref" href="rlsub_1.html#T79">RLSUB_1:79</a></i>;<br/>


</div>
