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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="matrlin.html#V2">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:25"/><i><font color="Green">E25</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>thus </b><a NAME="E2:25"/>
 <a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="pencil_1.html#V6">identifying_close_blocks</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D7">PENCIL_1:def 7</a><br/>

<b>assume </b><a NAME="E1:25_1"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2">PENCIL_1:2</a></i>;<br/>
<a NAME="E4:25_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E3:25_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:25_1"/><i><font color="Green">E28</font></i>: 
not  <a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="pencil_4.html#T18" target="_self">Th18</a>, <a class="txt" href="pencil_4.html#E1:25"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E6:25_1"/>
not  <a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="pencil_1.html#V3">void</a>
 
<b>by </b><i><a class="ref" href="pencil_4.html#T19" target="_self">Th19</a>, <a class="txt" href="pencil_4.html#E1:25"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E7:25_1"/><b>then </b>
not the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D4">PENCIL_1:def 4</a></i>;<br/>
<a NAME="E8:25_1"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="pencil_4.html#E4:25_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E10:25_1"/>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E5:25_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E12:25_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="vectsp_9.html#D3">VECTSP_9:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pencil_4.html#E12:25_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E14:25_1"/>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" target="_self">PencilSpace</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E5:25_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E16:25_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="vectsp_9.html#D3">VECTSP_9:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pencil_4.html#E16:25_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E19:25_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>16</sub></font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K3" target="_self">pencil</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="pencil_4.html#D4" target="_self">Def4</a>, <a class="txt" href="pencil_4.html#E8:25_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E21:25_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>18</sub></font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K3" target="_self">pencil</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="pencil_4.html#D4" target="_self">Def4</a>, <a class="txt" href="pencil_4.html#E8:25_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<i><font color="Green">E34</font></i>:  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>15</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="pencil_4.html#T12" target="_self">Th12</a>, <a class="txt" href="pencil_4.html#E1:25"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pencil_4.html#E19:25_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pencil_4.html#E4:25_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pencil_4.html#E3:25_1"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
 <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="pencil_4.html#T12" target="_self">Th12</a>, <a class="txt" href="pencil_4.html#E1:25"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pencil_4.html#E21:25_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="pencil_4.html#E4:25_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pencil_4.html#E3:25_1"><i><font color="Green">E26</font></i></a></i>
;<br/>
 <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="pencil_4.html#T12" target="_self">Th12</a>, <a class="txt" href="pencil_4.html#E1:25"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pencil_4.html#E19:25_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pencil_4.html#E4:25_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pencil_4.html#E3:25_1"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
 <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="pencil_4.html#T12" target="_self">Th12</a>, <a class="txt" href="pencil_4.html#E1:25"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pencil_4.html#E21:25_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="pencil_4.html#E4:25_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pencil_4.html#E3:25_1"><i><font color="Green">E26</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E24:25_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E22:25_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_4.html#E19:25_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pencil_4.html#E21:25_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pencil_4.html#T10" target="_self">Th10</a></i>;<br/>


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


</div>
