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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="E1:13"/><i><font color="Green">E14</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/>

<a NAME="E2:13"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="txt" href="pencil_4.html#E1:13"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</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="E4:13"/><i><font color="Green">E15</font></i>: 
 <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="vectsp_9.html#T39">VECTSP_9:39</a></i>;<br/>
<a NAME="E5:13"/>
<font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/>
<a NAME="E6:13"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E4:13"><i><font color="Green">E15</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</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>4</sub></font><b> such that </b><br/><a NAME="E8:13"/><i><font color="Green">E16</font></i>: 
 <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="ref" href="vectsp_9.html#T39">VECTSP_9:39</a></i>;<br/>
<a NAME="E9:13"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Maroon">c<sub>5</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>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E1:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="pencil_4.html#E8:13"><i><font color="Green">E16</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="vectsp_4.html#T34">VECTSP_4:34</a></i>;<br/>
<a NAME="E11:13"/>
 <a href="pencil_4.html#K3" target="_self">pencil</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="pencil_4.html#K4" target="_self">Pencils_of</a> <font color="Maroon">c<sub>2</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#E9:13"><i><font color="Green">E17</font></i></a>, <a class="txt" href="pencil_4.html#E4:13"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:13"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="pencil_4.html#K4" target="_self">Pencils_of</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


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