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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>thus </b><a NAME="E1:39"/>
( for <font color="Olive">b<sub>1</sub></font> being   <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> holds not the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>6</sub></font> being    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:39_1"/><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>assume </b><a NAME="E2:39_1"/>
( <font color="Maroon">c<sub>2</sub></font> is  not    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is  not    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E3:39_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T36">RLSUB_1:36</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:39_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E6:39_1"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:39_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E9:39_1"/><i><font color="Green">E38</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E3:39_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="rlsub_2.html#E5:39_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="rlsub_2.html#E8:39_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/>
<a NAME="E15:39_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="rlsub_1.html#V1">lineary-closed</a> )
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E1:39_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="rlsub_1.html#T42">RLSUB_1:42</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:39_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a></i>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/><a NAME="E3:39_1_1"/>
<font color="Maroon">c<sub>15</sub></font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T42">RLSUB_1:42</a></i>;<br/><a NAME="E4:39_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:39_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rlsub_1.html#T6">RLSUB_1:6</a></i>;<br/><a NAME="E5:39_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E6:39_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><b>hence </b><a NAME="E7:39_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rlsub_2.html#E9:39_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/><a NAME="E3:39_1_2"/>
<font color="Maroon">c<sub>15</sub></font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T42">RLSUB_1:42</a></i>;<br/><a NAME="E4:39_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:39_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rlsub_1.html#T6">RLSUB_1:6</a></i>;<br/><a NAME="E5:39_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E6:39_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><b>hence </b><a NAME="E7:39_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rlsub_2.html#E6:39_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:39_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rlsub_2.html#E1:39_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rlsub_2.html#E16:39_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rlsub_2.html#E17:39_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:39"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:39_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/><a NAME="E3:39_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>hence </b><a NAME="E4:39_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_3"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:39_3"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a></i>;<br/><a NAME="E3:39_3"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>hence </b><a NAME="E4:39_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:39"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E2:39"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rlsub_2.html#E3:39"><i><font color="Green">E34</font></i></a></i>;<br/>


</div>
