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<b>let </b><font color="Maroon" title="c1">GF</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="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c3">W</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="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>;<br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c4">A</font></span>)</span>
 ;<br/><a NAME="E2:18_1"/><b>then </b>
<font color="Maroon" title="c5">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">L</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font><b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12" title="VECTSP_7:th.12">VECTSP_7:12</a></i>;<br/><a NAME="E5:18_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c6">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/><a NAME="E6:18_1"/><b>then </b>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c6">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E1:18"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">K</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">W</font><b> such that </b><br/><a NAME="E8:18_1"/><i><font color="Green" title="E20">A3</font></i>: 
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c6">L</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c7">K</font> )
 <b>by </b><i><a class="ref" href="vectsp_9.html#T13" target="_self" title="VECTSP_9:th.13">Th13</a></i>;<br/><b>thus </b><a NAME="E9:18_1"/>
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E4:18_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="vectsp_9.html#E8:18_1"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:18"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:18"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c4">A</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T35" title="VECTSP_4:th.35">VECTSP_4:35</a></i>;<br/>


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