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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">A1</font>, <font color="Olive" title="b3">A2</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="b2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">A2</font> is <a href="lmod_5.html#V1" title="LMOD_5:attr.1">linearly-independent</a> &amp; <font color="Olive" title="b2">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">A2</font> holds <br/><span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Olive" title="b2">A1</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Olive" title="b3">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Olive" title="b1">V</font></span><br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">A2</font> is <a href="lmod_5.html#V1" title="LMOD_5:attr.1">linearly-independent</a> &amp; <font color="Olive" title="b1">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">A2</font> holds <br/><span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Olive" title="b2">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span><br/><b>let </b><font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">A2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">A2</font> is <a href="lmod_5.html#V1" title="LMOD_5:attr.1">linearly-independent</a> &amp; <font color="Maroon" title="c3">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A2</font> implies <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> )</span><br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c3">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">A2</font> is <a href="lmod_5.html#V1" title="LMOD_5:attr.1">linearly-independent</a> &amp; <font color="Maroon" title="c3">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">P</font> = <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A2</font></span>)</span> as  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> ;<br/>
<b>set </b><font color="Maroon" title="c6">Z</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P</font>;<br/>
<a NAME="E3:16"/><i><font color="Green" title="E7">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P</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 <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D2" title="VECTSP_5:def.2">VECTSP_5:def 2</a></i>;<br/>
<div><i><font color="Green" title="E8">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</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="c5">P</font> implies <font color="Maroon" title="c7">x</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="c2">V</font> )</span><br/><b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon" title="c7">x</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="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c2">V</font></span><br/><a NAME="E2:16_1"/><b>then </b>
( <font color="Maroon" title="c7">x</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="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A1</font></span>)</span> &amp; <font color="Maroon" title="c7">x</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="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="lmod_7.html#E3:16"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:16_1"/><b>then </b><i><font color="Green" title="E9">A4</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A2</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="c8">l1</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A1</font><b> such that </b><br/><a NAME="E5:16_1"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c7">x</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="c8">l1</font>
 <b>by </b><i><a class="ref" href="mod_3.html#T11" title="MOD_3:th.11">MOD_3:11</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">l2</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A2</font><b> such that </b><br/><a NAME="E7:16_1"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c7">x</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="c9">l2</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E3:16_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="mod_3.html#T11" title="MOD_3:th.11">MOD_3:11</a></i>;<br/><a NAME="E8:16_1"/><i><font color="Green" title="E12">A7</font></i>: 
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A1</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">l2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A2</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="E9:16_1"/><b>then </b><i><font color="Green" title="E13">A8</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">l2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">A2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></i>;<br/><a NAME="E10:16_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">l1</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c9">l2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">l2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#T74" title="VECTSP_6:th.74">VECTSP_6:74</a></i>;<br/><a NAME="E11:16_1"/><b>then </b>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">l1</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c9">l2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">A2</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E9:16_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">l</font> = <font color="Maroon" title="c8">l1</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c9">l2</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">A2</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 href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c10">l</font> = 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c8">l1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_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="c9">l2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T80" title="VECTSP_6:th.80">VECTSP_6:80</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E5:16_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="lmod_7.html#E7:16_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</a></i>
;<br/><a NAME="E14:16_1"/><b>then </b><i><font color="Green" title="E14">A9</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c10">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="lmod_7.html#E1:16"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="lmod_5.html#D1" title="LMOD_5:def.1">LMOD_5:def 1</a></i>;<br/><a NAME="E15:16_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_2"/><i><font color="Green" title="E15">A10</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c11">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l1</font>;<br/>
<b>consider </b><font color="Maroon" title="c12">b</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E4:16_1_2"/><i><font color="Green" title="E16">A11</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b</font> &amp; <font color="Maroon" title="c8">l1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</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> )
 <b>by </b><i><a class="txt" href="lmod_7.html#E1:16_1_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="vectsp_6.html#T19" title="VECTSP_6:th.19">VECTSP_6:19</a></i>;<br/>
<a NAME="E5:16_1_2"/>
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A1</font>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E8:16_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="lmod_7.html#E1:16_1_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="lmod_7.html#E4:16_1_2"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:16_1_2"/><b>then </b><i><font color="Green" title="E17">A12</font></i>: 
not <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A2</font>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E1:16"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> = 
<font color="Maroon" title="c10">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E14:16_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="vectsp_6.html#T20" title="VECTSP_6:th.20">VECTSP_6:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">l1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T73" title="VECTSP_6:th.73">VECTSP_6:73</a></i>
;<br/>
<b>then </b><font color="Maroon" title="c8">l1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font> = 
<font color="Maroon" title="c9">l2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T35" title="RLVECT_1:th.35">RLVECT_1:35</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E6:16_1_2"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="lmod_7.html#T2" target="_self" title="LMOD_7:th.2">Th2</a></i>
;<br/>
<b>hence </b><a NAME="E9:16_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="lmod_7.html#E4:16_1_2"><i><font color="Green" title="E16">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E16:16_1"/>
<font color="Maroon" title="c7">x</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="c2">V</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E5:16_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="vectsp_6.html#T45" title="VECTSP_6:th.45">VECTSP_6:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E5:16"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T25" title="VECTSP_4:th.25">VECTSP_4:25</a></i>;<br/>
<a NAME="E6:16"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</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="c5">P</font> iff <font color="Olive" title="b1">x</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="c2">V</font> )
 
<b>by </b><i><a class="txt" href="lmod_7.html#E4:16"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E7:16"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:16"/>
<span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c4">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
