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<span class="kw">let </span><font color="Maroon" title="c1">K</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">KL1</font>, <font color="Olive" title="b4">KL2</font> being   <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b5">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b5">X</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b3">KL1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">X</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b4">KL2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">X</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b3">KL1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b4">KL2</font></span>)</span> holds <br/><font color="Olive" title="b3">KL1</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Olive" title="b4">KL2</font></span><br/><span class="kw">let </span><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">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">KL1</font>, <font color="Olive" title="b3">KL2</font> being   <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b4">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b4">X</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b2">KL1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">X</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b3">KL2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">X</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b2">KL1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b3">KL2</font></span>)</span> holds <br/><font color="Olive" title="b2">KL1</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Olive" title="b3">KL2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">KL1</font>, <font color="Olive" title="b2">KL2</font> being   <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b3">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b3">X</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b1">KL1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">X</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b2">KL2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">X</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">KL1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b2">KL2</font></span>)</span> holds <br/><font color="Olive" title="b1">KL1</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Olive" title="b2">KL2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">KL1</font>, <font color="Maroon" title="c5">KL2</font> be    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">X</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c4">KL1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">KL2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c4">KL1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c5">KL2</font></span>)</span> holds <br/><font color="Maroon" title="c4">KL1</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">KL2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">X</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c4">KL1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">KL2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c4">KL1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c5">KL2</font></span>)</span> implies <font color="Maroon" title="c4">KL1</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">KL2</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
( <font color="Maroon" title="c6">X</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c4">KL1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> )
 <span class="kw">and </span><br/><a NAME="E2:13"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">KL2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c4">KL1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c5">KL2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">KL1</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">KL2</font></span><br/>

<a NAME="E3:13"/>
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">KL2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c4">KL1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">KL2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="vectsp_6.html#T59" title="VECTSP_6:th.59">VECTSP_6:59</a>, <a class="ref" href="vectsp_6.html#T78" title="VECTSP_6:th.78">VECTSP_6:78</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:13"/>
<font color="Maroon" title="c4">KL1</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">KL2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="matrlin.html#T9" target="_self" title="MATRLIN:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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