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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">KL1</font> be    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">KL2</font> be    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 ;<br/>

<b>assume </b><a NAME="E2:1"/><i><font color="Green">E32</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">KL1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/>

<b>assume </b><a NAME="E3:1"/><i><font color="Green">E33</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">KL2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/>

<b>assume </b><a NAME="E4:1"/><i><font color="Green">E34</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">KL1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">KL2</font>
 ;<br/>

<a NAME="E5:1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">KL1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">KL2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E6:1"/>
 <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">KL2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">KL1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">KL2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T80">RLVECT_2:80</a></i>;<br/>
<a NAME="E7:1"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">KL2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">KL2</font></span>)</span> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<b>by </b><i>, <a class="ref" href="rlvect_3.html#T6">RLVECT_3:6</a></i>;<br/>
<a NAME="E9:1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">KL1</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">KL2</font> is    <a href="rlvect_2.html#M2">Linear_Combination</a> of  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">KL2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">KL1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">KL2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">KL1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">KL1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
;<br/>
<a NAME="E11:1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">KL2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="rlvect_3.html#T4">RLVECT_3:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><a NAME="E1:1_2"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">KL2</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="rlvect_3.html#D1">RLVECT_3:def 1</a></i>;<br/><a NAME="E2:1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">KL2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/><a NAME="E3:1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">KL2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rlvect_2.html#T79">RLVECT_2:79</a></i>;<br/><b>hence </b><a NAME="E4:1_2"/>
<font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:1"/>
<font color="Maroon">KL1</font> <a href="rlvect_2.html#R1">=</a> <font color="Maroon">KL2</font>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D11">RLVECT_2:def 11</a></i>;<br/>


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