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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>  st  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b1">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> <font color="Olive" title="b2">W2</font> holds <br/> for <font color="Olive" title="b3">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b3">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Olive" title="b1">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Olive" title="b2">W2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c3">W2</font> be   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">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> <font color="Maroon" title="c3">W2</font> implies  for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c2">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c3">W2</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:26"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">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> <font color="Maroon" title="c3">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c2">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c3">W2</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c2">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c3">W2</font></span><br/>

<a NAME="E2:26"/>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="vectsp_4.html#T24" title="VECTSP_4:th.24">VECTSP_4:24</a></span>;<br/>
<a NAME="E3:26"/><span class="kw">then </span>
<font color="Maroon" title="c2">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_4.html#T5" target="_self" title="PENCIL_4:th.5">Th6</a></span>;<br/>
<a NAME="E4:26"/><span class="kw">then </span>
<font color="Maroon" title="c2">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_4.html#T3" target="_self" title="PENCIL_4:th.3">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:26"/>
<font color="Maroon" title="c4">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c2">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c3">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T38" title="VECTSP_9:th.38">VECTSP_9:38</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c3">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c2">W1</font></span><br/>

<a NAME="E6:26"/>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W2</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="vectsp_4.html#T24" title="VECTSP_4:th.24">VECTSP_4:24</a></span>;<br/>
<a NAME="E7:26"/><span class="kw">then </span>
<font color="Maroon" title="c3">W2</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_4.html#T5" target="_self" title="PENCIL_4:th.5">Th6</a></span>;<br/>
<a NAME="E8:26"/><span class="kw">then </span>
<font color="Maroon" title="c3">W2</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_4.html#T3" target="_self" title="PENCIL_4:th.3">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:26"/>
<font color="Maroon" title="c4">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c3">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c2">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T38" title="VECTSP_9:th.38">VECTSP_9:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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