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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">L</font> be    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>;<br/>







<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E11">A1</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c4">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>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">C</font> =  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c4">L</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">W</font> ;<br/>
<a NAME="E3:11"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</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="c2">V</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">K</font> = <font color="Maroon" title="c4">L</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font> <b>by </b><i><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></i>;<br/>
<a NAME="E5:11"/><i><font color="Green" title="E12">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green" title="E13">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">w</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W</font>;<br/><b>assume </b><a NAME="E1:11_1"/>
not <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font>
 ;<br/><a NAME="E2:11_1"/><b>then </b>
( not <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c7">w</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></i>;<br/><a NAME="E3:11_1"/><b>then </b>
<font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">GF</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#T20" title="VECTSP_6:th.20">VECTSP_6:20</a></i>;<br/><b>hence </b><a NAME="E4:11_1"/>
<font color="Maroon" title="c6">K</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">GF</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E5:11"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:11"/>
<font color="Maroon" title="c6">K</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">K</font> = <font color="Maroon" title="c6">K</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">W</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E6:11"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="vectsp_6.html#D4" title="VECTSP_6:def.4">VECTSP_6:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">K</font>
;<br/>

<b>thus </b><a NAME="E9:11"/>
(  <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="c4">L</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c7">K</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="c4">L</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E1:11"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="vectsp_9.html#T11" target="_self" title="VECTSP_9:th.11">Th11</a></i>;<br/>


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