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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="rusub_4.html#V1" target="_self">finite-dimensional</a> <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="bhsp_1.html#NM1">RealUnitarySpace</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/>
<a NAME="E3:24"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font> is <a href="rusub_4.html#V1" target="_self">finite-dimensional</a>
 
<b>by </b><i><a class="ref" href="rusub_4.html#T7" target="_self">Th7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:24"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="rusub_4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E6:24"/><i><font color="Green">E19</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rusub_4.html#K1" target="_self">dim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E3:24"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rusub_4.html#E5:24"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rusub_4.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E7:24"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="rusub_2.html#T16">RUSUB_2:16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E9:24"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E5:24"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rusub_3.html#T24">RUSUB_3:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="ref" href="rusub_4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E11:24"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="rusub_2.html#T16">RUSUB_2:16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E13:24"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E5:24"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rusub_3.html#T24">RUSUB_3:24</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="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="ref" href="rusub_4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E15:24"/><i><font color="Green">E22</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="rusub_4.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E16:24"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> is <a href="rusub_4.html#V1" target="_self">finite-dimensional</a>
 
<b>by </b><i><a class="ref" href="rusub_4.html#T7" target="_self">Th7</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:24_1"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:24_1"/><b>then </b><i><font color="Green">E25</font></i>: 
( <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>5</sub></font> or <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> )
 ;<br/><a NAME="E4:24_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="rusub_4.html#E3:24_1"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E6:24_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_4.html#E2:24_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E7:24_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="rusub_2.html#T2">RUSUB_2:2</a></i>;<br/><b>hence </b><a NAME="E8:24_1"/>
<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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E19:24"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E9:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="rusub_4.html#E13:24"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_2"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E4:24_2"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:24_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_4.html#E3:24_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:24_2"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="rusub_3.html#T2">RUSUB_3:2</a></i>;<br/><a NAME="E7:24_2"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>5</sub></font> #) &amp; <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>6</sub></font> #) )
 <b>by </b><i><a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/><a NAME="E8:24_2"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>14</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="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E9:24_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K3">/\</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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:24_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rusub_2.html#D2">RUSUB_2:def 2</a></i>;<br/><a NAME="E11:24_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,the <a href="bhsp_1.html#U1">scalar</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> #)
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E12:24_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E5:24"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/><a NAME="E13:24_2"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="rusub_4.html#E8:24_2"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E15:24_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:24_2_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:24_2_1"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:24_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/><a NAME="E4:24_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:24_2_1"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_4.html#E2:24_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:24_2_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E8:24_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="rusub_4.html#E13:24_2"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E19:24_2"/>
( <font color="Maroon">c<sub>9</sub></font> is <a href="rlvect_3.html#V1">linearly-independent</a> &amp; <font color="Maroon">c<sub>9</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="rlvect_3.html#V1">linearly-independent</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="rusub_3.html#T22">RUSUB_3:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:24_2_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:24_2_2"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:24_2_2"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_4.html#E9:24"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:24_2_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E3:24_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E22:24_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E23:24_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>17</sub></font> 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="E24:24_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E25:24_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> = 
<font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="rusub_4.html#E4:24_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="rusub_4.html#E3"><i><font color="Green">Lemma3</font></i></a></i>
;<br/><a NAME="E27:24_2"/><b>then </b>
not <font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E23:24_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rusub_4.html#E25:24_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rusub_3.html#T25">RUSUB_3:25</a></i>;<br/><b>hence </b><a NAME="E28:24_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rusub_4.html#E12:24_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rusub_3.html#T28">RUSUB_3:28</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:24"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E19:24"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E22:24"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T64">CARD_2:64</a></i>;<br/>
<a NAME="E23:24"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E25:24"/><i><font color="Green">E26</font></i>: 
 <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="rusub_3.html#T28">RUSUB_3:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:24_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><a NAME="E2:24_3"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E4:24_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E5:24_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E6:24_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="rusub_2.html#T1">RUSUB_2:1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="rusub_4.html#E4:24_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="rusub_4.html#E5:24_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E9:24_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T21">RUSUB_3:21</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E11:24_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T1">RUSUB_3:1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E13:24_3"/><i><font color="Green">E31</font></i>: 
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="ref" href="rusub_3.html#T19">RUSUB_3:19</a></i>;<br/><a NAME="E14:24_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T21">RUSUB_3:21</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>11</sub></font><b> such that </b><br/><a NAME="E16:24_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T1">RUSUB_3:1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E18:24_3"/><i><font color="Green">E33</font></i>: 
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>22</sub></font> &amp;  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="ref" href="rusub_3.html#T19">RUSUB_3:19</a></i>;<br/><b>set </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>23</sub></font>;<br/><a NAME="E19:24_3"/>
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_4.html#E13:24_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rusub_4.html#E18:24_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><a NAME="E20:24_3"/><b>then </b>
(  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>23</sub></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">c<sub>21</sub></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">c<sub>23</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>21</sub></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">c<sub>23</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="rlvect_2.html#T58">RLVECT_2:58</a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/><a NAME="E21:24_3"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>23</sub></font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><a NAME="E23:24_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E6:24_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rusub_4.html#E11:24_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rusub_4.html#E13:24_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rusub_4.html#E16:24_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rusub_4.html#E18:24_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="rlvect_3.html#T1">RLVECT_3:1</a></i>;<br/><a NAME="E24:24_3"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T1">RUSUB_3:1</a></i>;<br/><b>hence </b><a NAME="E25:24_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:24"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E28:24"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="rusub_1.html#M1">Subspace</a> of  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#T22">RUSUB_1:22</a></i>;<br/>
<a NAME="E29:24"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E25:24"><i><font color="Green">E26</font></i></a>, <a class="ref" href="rusub_1.html#T20">RUSUB_1:20</a></i>;<br/>
<a NAME="E30:24"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>13</sub></font> st  <a href="rlvect_2.html#K6">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/> <a href="rlvect_2.html#K3">Carrier</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>13</sub></font>;<br/>

<b>assume </b><a NAME="E1:24_4"/><i><font color="Green">E28</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:24_4"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="rusub_2.html#T7">RUSUB_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="ref" href="rusub_2.html#T7">RUSUB_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="ref" href="rusub_2.html#T7">RUSUB_2:7</a></i>;<br/>
<a NAME="E5:24_4"/><i><font color="Green">E30</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:24_4"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>18</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E8:24_4"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E9:24_4"/><i><font color="Green">E33</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D10">RLVECT_2:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <b>by </b><i><a class="txt" href="rusub_4.html#E8:24_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font><b> such that </b><br/><a NAME="E12:24_4"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="finseq_3.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="finseq_3.html#K1">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E7:24_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_3.html#T124">FINSEQ_3:124</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="finseq_3.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="finseq_3.html#K1">-</a> <font color="Maroon">c<sub>20</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <b>by </b><i><a class="ref" href="finseq_3.html#T93">FINSEQ_3:93</a></i>;<br/>
<a NAME="E14:24_4"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>23</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="rusub_4.html#E7:24_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_3.html#T94">FINSEQ_3:94</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E16:24_4"/><i><font color="Green">E36</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E17:24_4"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_5.html#T8">RLVECT_5:8</a></i>;<br/>
<a NAME="E18:24_4"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E16:24_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E19:24_4"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E14:24_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="rusub_4.html#E17:24_4"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rlvect_5.html#T7">RLVECT_5:7</a></i>;<br/>
<a NAME="E20:24_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> ;<br/>
<font color="Maroon">c<sub>25</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">c<sub>25</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E23:24_4"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>;<br/>
<b>then </b><i><font color="Green">E39</font></i>:  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rusub_4.html#E16:24_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>9</sub></font>

;<br/>
<a NAME="E25:24_4"/><b>then </b><i><font color="Green">E40</font></i>: 
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E26:24_4"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E28:24_4"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E29:24_4"/><i><font color="Green">E41</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T20">RUSUB_3:20</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E31:24_4"/><i><font color="Green">E42</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E32:24_4"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_5.html#T8">RLVECT_5:8</a></i>;<br/>
<a NAME="E33:24_4"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E31:24_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E34:24_4"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E14:24_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="rusub_4.html#E32:24_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rlvect_5.html#T7">RLVECT_5:7</a></i>;<br/>
<a NAME="E35:24_4"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_4.html#E8:24_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a></i>
;<br/>
<a NAME="E37:24_4"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E31:24_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="rusub_4.html#E35:24_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E38:24_4"/><b>then </b>
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</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="txt" href="rusub_4.html#E5:24_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/>
<a NAME="E39:24_4"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>28</sub></font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>17</sub></font><b> such that </b><br/><a NAME="E41:24_4"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E42:24_4"/><i><font color="Green">E47</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T20">RUSUB_3:20</a></i>;<br/>
<i><font color="Green">E48</font></i>:  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_2.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rusub_4.html#E1:24_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="rusub_4.html#E9:24_4"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rusub_4.html#E12:24_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rlvect_5.html#T5">RLVECT_5:5</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_3.html#T41">RLVECT_3:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_4.html#E19:24_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="rusub_4.html#E34:24_4"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
;<br/>
<b>then </b> <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>24</sub></font> = 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D10">RLVECT_1:def 10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_4.html#E42:24_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rusub_1.html#T9">RUSUB_1:9</a></i>
;<br/>
<a NAME="E45:24_4"/><b>then </b>
(  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>26</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>26</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="rusub_4.html#E29:24_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E46:24_4"/><b>then </b>
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>26</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="rusub_2.html#T3">RUSUB_2:3</a></i>;<br/>
<a NAME="E47:24_4"/><b>then </b>
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>26</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E5:24"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>29</sub></font> being    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E49:24_4"/><i><font color="Green">E49</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T1">RUSUB_3:1</a></i>;<br/>
<a NAME="E50:24_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="rusub_2.html#T22">RUSUB_2:22</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>30</sub></font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E52:24_4"/><i><font color="Green">E50</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>29</sub></font>
 <b>and </b><br/><a NAME="E53:24_4"/><i><font color="Green">E51</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T19">RUSUB_3:19</a></i>;<br/>
<a NAME="E54:24_4"/><i><font color="Green">E52</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>27</sub></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">c<sub>30</sub></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">c<sub>27</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T58">RLVECT_2:58</a></i>;<br/>
<a NAME="E55:24_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E13:24"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E56:24_4"/>
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>30</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="rusub_4.html#E5:24_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rusub_4.html#E37:24_4"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rusub_4.html#E52:24_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a>, <a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/>
<a NAME="E57:24_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>30</sub></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">c<sub>27</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E55:24_4"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E58:24_4"/><b>then </b><i><font color="Green">E54</font></i>: 
(  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</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="txt" href="rusub_4.html#E54:24_4"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E59:24_4"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="tarski.html#R1">c=</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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>31</sub></font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E61:24_4"/><i><font color="Green">E55</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E62:24_4"/><i><font color="Green">E56</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rusub_4.html#E2:24_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rusub_3.html#T20">RUSUB_3:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>31</sub></font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="rusub_4.html#E58:24_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="rusub_4.html#E61:24_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<a NAME="E64:24_4"/>
( <font color="Maroon">c<sub>9</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="rlvect_3.html#V1">linearly-independent</a> )
 
<b>by </b><i><a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="rlvect_3.html#V1">linearly-independent</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <b>by </b><i><a class="txt" href="rusub_4.html#E2:24_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rusub_3.html#T22">RUSUB_3:22</a></i>;<br/>
<a NAME="E66:24_4"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>30</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E52:24_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<a NAME="E67:24_4"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>30</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E9:24"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E68:24_4"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="rlvect_2.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E25:24_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="rusub_4.html#E29:24_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rusub_4.html#E49:24_4"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rusub_4.html#E53:24_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rlvect_5.html#T1">RLVECT_5:1</a></i>;<br/>
<a NAME="E69:24_4"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_4.html#E29:24_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="rusub_4.html#E43:24_4"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rusub_4.html#E49:24_4"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rusub_4.html#E53:24_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rusub_1.html#T5">RUSUB_1:5</a></i>
<br/>.= 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="rusub_4.html#E62:24_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="rlvect_3.html#T1">RLVECT_3:1</a></i>
;<br/>
<a NAME="E71:24_4"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E61:24_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rusub_4.html#E68:24_4"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rusub_4.html#E69:24_4"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rlvect_3.html#D1">RLVECT_3:def 1</a></i>;<br/>
<a NAME="E72:24_4"/><i><font color="Green">E60</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></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">c<sub>27</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E24:24_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rusub_4.html#E37:24_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_1.html#T51">XBOOLE_1:51</a></i>;<br/>
<a NAME="E73:24_4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rusub_4.html#E24:24_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rusub_4.html#E37:24_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="rusub_4.html#E73:24_4"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/>
<a NAME="E75:24_4"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_4_8"/>
not  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>34</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_4_8"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E4:24_4_8"/><i><font color="Green">E64</font></i>: 
not <font color="Maroon">c<sub>34</sub></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">c<sub>24</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>34</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <b>by </b><i><a class="txt" href="rusub_4.html#E3:24_4_8"><i><font color="Green">E63</font></i></a></i>;<br/><i><font color="Green">E65</font></i>: 0 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>35</sub></font>
<b>by </b><i><a class="txt" href="rusub_4.html#E4:24_4_8"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#D12">RLVECT_2:def 12</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_4_8_2"/>
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></font> or <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font> )
 </span><b>by </b><i><a class="txt" href="rusub_4.html#E72:24_4"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rusub_4.html#E3:24_4_8"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4_8_2_1"><b>suppose </b></a><a NAME="E1:24_4_8_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>36</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:24_4_8_2_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="rlvect_5.html#T3">RLVECT_5:3</a></i>;<br/><a NAME="E4:24_4_8_2_1"/>
not <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E75:24_4"><i><font color="Green">E62</font></i></a>, <a class="txt" href="rusub_4.html#E1:24_4_8_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:24_4_8_2_1"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>35</sub></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/><b>hence </b><a NAME="E6:24_4_8_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rusub_4.html#E6:24_4_8"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rusub_4.html#E3:24_4_8_2_1"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4_8_2_2"><b>suppose </b></a><a NAME="E1:24_4_8_2_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>36</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:24_4_8_2_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="rlvect_5.html#T3">RLVECT_5:3</a></i>;<br/><a NAME="E4:24_4_8_2_2"/>
not <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E75:24_4"><i><font color="Green">E62</font></i></a>, <a class="txt" href="rusub_4.html#E1:24_4_8_2_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:24_4_8_2_2"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>35</sub></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/><b>hence </b><a NAME="E6:24_4_8_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rusub_4.html#E6:24_4_8"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rusub_4.html#E3:24_4_8_2_2"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E77:24_4"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="rusub_4.html#E71:24_4"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:24"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="rlvect_3.html#D1">RLVECT_3:def 1</a></i>;<br/>
<a NAME="E32:24"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> is    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E29:24"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="rusub_4.html#K1" target="_self">dim</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K1" target="_self">dim</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_4.html#E6:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="rusub_4.html#E16:24"><i><font color="Green">E23</font></i></a>, <a class="txt" href="rusub_4.html#E22:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rusub_4.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_4.html#E15:24"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rusub_4.html#D3" target="_self">Def3</a></i>
;<br/>
<b>hence </b><a NAME="E34:24"/>
<span class="p1">(<span class="default"><a href="rusub_4.html#K1" target="_self">dim</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K1" target="_self">dim</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rusub_2.html#K2">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>


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