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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>;<br/><b>let </b><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font>, <font color="Maroon" title="c5">P0</font>, <font color="Maroon" title="c6">Q0</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>







<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E13">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c5">P0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c6">Q0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c5">P0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">Q0</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">P</font> being  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:14"/><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P0</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="vectsp_9.html#D3" title="VECTSP_9:def.3">VECTSP_9:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">Q</font> being  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E5:14"/><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c9">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q0</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c9">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="vectsp_9.html#D3" title="VECTSP_9:def.3">VECTSP_9:def 3</a></i>;<br/>
<a NAME="E6:14"/>
<font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c8">P</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T20" title="VECTSP_5:th.20">VECTSP_5:20</a></i>;<br/>
<a NAME="E7:14"/><b>then </b><i><font color="Green" title="E16">A4</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E3:14"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="vectsp_9.html#T29" title="VECTSP_9:th.29">VECTSP_9:29</a></i>;<br/>
<a NAME="E8:14"/>
( <font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c8">P</font> &amp; <font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c9">Q</font> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E3:14"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="pencil_4.html#E5:14"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="pencil_4.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="pencil_4.html#T9" target="_self" title="PENCIL_4:th.9">Th9</a></i>;<br/>
<a NAME="E9:14"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T24" title="VECTSP_5:th.24">VECTSP_5:24</a></i>;<br/>
<a NAME="E10:14"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T29" title="VECTSP_9:th.29">VECTSP_9:29</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1"/>
(  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font></span>)</span> or  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> )
 </span><b>by </b><i><a class="txt" href="pencil_4.html#E10:14"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="pencil_4.html#E7:14"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</a>, <a class="txt" href="pencil_4.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>suppose </b></a><a NAME="E1:14_1_1"/><i><font color="Green" title="E19">A7</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:14_1_1"/><b>then </b>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E9:14"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="vectsp_9.html#T32" title="VECTSP_9:th.32">VECTSP_9:32</a></i>;<br/><b>hence </b><a NAME="E3:14_1_1"/>
<font color="Maroon" title="c5">P0</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c6">Q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W1</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:14"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="pencil_4.html#E5:14"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/><a NAME="E4:14_1_1"/>
( <font color="Maroon" title="c8">P</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c9">Q</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">W2</font> )
 <b>by </b><i><a class="ref" href="pencil_4.html#T9" target="_self" title="PENCIL_4:th.9">Th9</a>, <a class="txt" href="pencil_4.html#E3:14"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="pencil_4.html#E5:14"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="pencil_4.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/><a NAME="E5:14_1_1"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c9">Q</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T40" title="VECTSP_5:th.40">VECTSP_5:40</a></i>;<br/><a NAME="E6:14_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c9">Q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c9">Q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W1</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:14_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="vectsp_9.html#T36" title="VECTSP_9:th.36">VECTSP_9:36</a></i>;<br/><a NAME="E7:14_1_1"/><b>then </b>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c9">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="pencil_4.html#E3:14"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="pencil_4.html#E5:14"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="pencil_4.html#E5:14_1_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="vectsp_9.html#T32" title="VECTSP_9:th.32">VECTSP_9:32</a></i>;<br/><b>hence </b><a NAME="E8:14_1_1"/>
<font color="Maroon" title="c5">P0</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c6">Q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:14"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="pencil_4.html#E5:14"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><b>suppose </b></a><a NAME="E1:14_1_2"/><i><font color="Green" title="E19">A9</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ;<br/><div class="add"><a NAME="E2:14_1_2"/>
<font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c8">P</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T20" title="VECTSP_5:th.20">VECTSP_5:20</a></i>;<br/><a NAME="E3:14_1_2"/><b>then </b><i><font color="Green" title="E20">A10</font></i>: 
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c8">P</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T32" title="VECTSP_9:th.32">VECTSP_9:32</a>, <a class="txt" href="pencil_4.html#E1:14_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="pencil_4.html#E3:14"><i><font color="Green" title="E14">A2</font></i></a></i>;<br/><a NAME="E4:14_1_2"/>
<font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c9">Q</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T20" title="VECTSP_5:th.20">VECTSP_5:20</a></i>;<br/><a NAME="E5:14_1_2"/><b>then </b>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c9">Q</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T32" title="VECTSP_9:th.32">VECTSP_9:32</a>, <a class="txt" href="pencil_4.html#E1:14_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="pencil_4.html#E5:14"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E6:14_1_2"/>
( <font color="Maroon" title="c5">P0</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c6">Q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Maroon" title="c5">P0</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c6">Q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c4">W2</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="pencil_4.html#E3:14"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="pencil_4.html#E5:14"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="pencil_4.html#E3:14_1_2"><i><font color="Green" title="E20">A10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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