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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <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="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:15"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E2:15"/><b>then </b>
(  <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="rusub_1.html#M1">Subspace</a> of  <a href="rusub_3.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="rusub_1.html#M1">Subspace</a> of  <a href="rusub_3.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rusub_3.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E3:15"/><b>then </b><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rusub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is    <a href="rusub_1.html#M1">Subspace</a> of  <a href="rusub_3.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rusub_3.html#E12"><i><font color="Green">Lemma9</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T1" target="_self">Th1</a></i>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font>;<br/><div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1"/>
<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>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:15_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><b>hence </b><a NAME="E4:15_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  )
 ;<br/><b>assume </b><a NAME="E5:15_1_1"/>
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>thus </b><a NAME="E6:15_1_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 0;<br/><a NAME="E5:15_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) )
 <b>by </b><i><a class="txt" href="rusub_3.html#E4:15_1"><i><font color="Green">E13</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E7:15_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5">FUNCT_2:sch 5</a>(<a class="txt" href="rusub_3.html#E5:15_1"><i><font color="Green">E14</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then 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> ;<br/><a NAME="E3:15_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><b>hence </b><a NAME="E4:15_1_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  )
 ;<br/><b>assume </b><a NAME="E5:15_1_2"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>thus </b><a NAME="E6:15_1_2"/>
0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E10:15_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  ) )
 <b>by </b><i><a class="txt" href="rusub_3.html#E9:15_1"><i><font color="Green">E16</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E12:15_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5">FUNCT_2:sch 5</a>(<a class="txt" href="rusub_3.html#E10:15_1"><i><font color="Green">E17</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</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>1</sub></font> ;<br/><a NAME="E15:15_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rusub_3.html#E7:15_1"><i><font color="Green">E15</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlvect_2.html#D5">RLVECT_2:def 5</a></i>;<br/><a NAME="E17:15_1"/><i><font color="Green">E19</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_1_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:15_1_3"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 
;<br/>
<a NAME="E3:15_1_3"/><b>then </b><i><font color="Green">E20</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
;<br/>
<b>assume </b><a NAME="E4:15_1_3"/>
not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>hence </b><a NAME="E5:15_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="rusub_3.html#E7:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="rusub_3.html#E3:15_1_3"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:15_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E19:15_1"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rusub_3.html#E17:15_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</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_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</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>1</sub></font> ;<br/><a NAME="E22:15_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rusub_3.html#E12:15_1"><i><font color="Green">E18</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlvect_2.html#D5">RLVECT_2:def 5</a></i>;<br/><a NAME="E24:15_1"/><i><font color="Green">E20</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_1_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E2:15_1_4"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 
;<br/>
<a NAME="E3:15_1_4"/><b>then </b><i><font color="Green">E21</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
;<br/>
<b>assume </b><a NAME="E4:15_1_4"/>
not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>hence </b><a NAME="E5:15_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="rusub_3.html#E12:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rusub_3.html#E3:15_1_4"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:15_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_1_5"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:15_1_5"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_5"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:15_1"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rusub_3.html#E24:15_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><a NAME="E28:15_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_2.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>17</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_2.html#D11">RLVECT_2:def 11</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_6_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> or  not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_6_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_6_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_6_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:15_1_6_1_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:15_1_6_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rusub_3.html#E1:15_1_6_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#D12">RLVECT_2:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_3.html#E7:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="rusub_3.html#E1:15_1_6_1_1_1"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="txt" href="rusub_3.html#E12:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rusub_3.html#E2:15_1_6_1_1_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_6_1_1_2"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_6_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_6_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_6_1_1_2_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font> or  not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_6_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_6_1_1_2_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_6_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_6_1_1_2_1_1_1_1"/>
not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:15_1_6_1_1_2_1_1_1_1"/><b>then </b>
(  not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:15_1_6_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rusub_3.html#E1:15_1_6_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rusub_3.html#E1:15_1_6_1_1_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#D12">RLVECT_2:def 12</a></i>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_3.html#E7:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="rusub_3.html#E1:15_1_6_1_1_2"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="rusub_3.html#E12:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rusub_3.html#E2:15_1_6_1_1_2_1_1_1"><i><font color="Green">E23</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_6_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_6_1_1_2_1_1_2"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:15_1_6_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
( not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#D12">RLVECT_2:def 12</a></i>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_3.html#E7:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="rusub_3.html#E2:15_1_6_1_1_2_1_1_2"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> 0
<b>by </b><i><a class="txt" href="rusub_3.html#E12:15_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rusub_3.html#E2:15_1_6_1_1_2_1_1_2"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="rusub_3.html#E1:15_1_6_1_1_2_1_1_2"><i><font color="Green">E22</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="E3:15_1_6_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:15_1_6"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E29:15_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>14</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>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rusub_3.html#E3:15_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="rlvect_3.html#T1">RLVECT_3:1</a></i>;<br/><a NAME="E30:15_1"/>
(  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="rusub_3.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E31:15_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rusub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rusub_3.html#E29:15_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rusub_2.html#T1">RUSUB_2:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:15"/><b>then </b>
 <a href="rusub_3.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is    <a href="rusub_1.html#M1">Subspace</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rusub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rusub_1.html#T23">RUSUB_1:23</a></i>;<br/>
<b>hence </b><a NAME="E6:15"/>
 <a href="rusub_3.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rusub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rusub_3.html#E3:15"><i><font color="Green">E11</font></i></a>, <a class="ref" href="rusub_1.html#T20">RUSUB_1:20</a></i>;<br/>


</div>
