<?xml version="1.0"?>
<div class="add">

<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="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <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">W1</font>, <font color="Maroon" title="c4">W2</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>reconsider </b><font color="Maroon" title="c5">V</font> = <font color="Maroon" title="c2">V</font> as   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">GF</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">W1</font> = <font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c7">W2</font> = <font color="Maroon" title="c4">W2</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c5">V</font> ;<br/>
<a NAME="E3:39"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T28" target="_self" title="VECTSP_9:th.28">Th28</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">I</font> being  <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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font></span>)</span><b> such that </b><br/><a NAME="E5:39"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c8">I</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D1" target="_self" title="VECTSP_9:def.1">Def1</a></i>;<br/>
<a NAME="E6:39"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E3:39"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="vectsp_9.html#E5:39"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="vectsp_9.html#D2" target="_self" title="VECTSP_9:def.2">Def2</a></i>;<br/>
<a NAME="E7:39"/>
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W1</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T20" title="VECTSP_5:th.20">VECTSP_5:20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">I1</font> being    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c6">W1</font><b> such that </b><br/><a NAME="E9:39"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c8">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">I1</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E5:39"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="vectsp_9.html#T17" target="_self" title="VECTSP_9:th.17">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">I1</font> = <font color="Maroon" title="c9">I1</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="c6">W1</font> <b>by </b><i><a class="ref" href="vectsp_9.html#T24" target="_self" title="VECTSP_9:th.24">Th24</a></i>;<br/>
<a NAME="E11:39"/>
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c7">W2</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T20" title="VECTSP_5:th.20">VECTSP_5:20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">I2</font> being    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c7">W2</font><b> such that </b><br/><a NAME="E13:39"/><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon" title="c8">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">I2</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E5:39"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="vectsp_9.html#T17" target="_self" title="VECTSP_9:th.17">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">I2</font> = <font color="Maroon" title="c11">I2</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="c7">W2</font> <b>by </b><i><a class="ref" href="vectsp_9.html#T24" target="_self" title="VECTSP_9:th.24">Th24</a></i>;<br/>
<a NAME="E15:39"/><i><font color="Green" title="E37">A6</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c12">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W2</font>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#D2" target="_self" title="VECTSP_9:def.2">Def2</a></i>;<br/>
<a NAME="E16:39"/><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T28" target="_self" title="VECTSP_9:th.28">Th28</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">A</font> = <font color="Maroon" title="c10">I1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">I2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">I1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">I2</font>
 ;<br/><a NAME="E2:39_1"/><b>then </b><i><font color="Green" title="E39">A8</font></i>: 
( <font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">I1</font> or <font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">I2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:39_1"/><b>then </b><i><font color="Green" title="E40">A9</font></i>: 
( <font color="Maroon" title="c14">v</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="c6">W1</font> or <font color="Maroon" title="c14">v</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="c7">W2</font> )
 ;<br/><a NAME="E4:39_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">W1</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="c5">V</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W2</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="c5">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="c15">v'</font> = <font color="Maroon" title="c14">v</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c5">V</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E3:39_1"><i><font color="Green" title="E40">A9</font></i></a></i>;<br/><a NAME="E6:39_1"/>
( <font color="Maroon" title="c15">v'</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> or <font color="Maroon" title="c15">v'</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W2</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E2:39_1"><i><font color="Green" title="E39">A8</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="E7:39_1"/><b>then </b>
<font color="Maroon" title="c15">v'</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T6" title="VECTSP_5:th.6">VECTSP_5:6</a></i>;<br/><b>hence </b><a NAME="E8:39_1"/>
<font color="Maroon" title="c14">v</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"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">A</font> = <font color="Maroon" title="c10">I1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">I2</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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E19:39"/><i><font color="Green" title="E39">A10</font></i>: 
<font color="Maroon" title="c8">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">I1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">I2</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E9:39"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="vectsp_9.html#E13:39"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/>
not <font color="Maroon" title="c10">I1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">I2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">I</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_2"/><i><font color="Green" title="E40">A11</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">I1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">I2</font>
 <b>and </b><br/><a NAME="E4:39_2"/><i><font color="Green" title="E41">A12</font></i>: 
not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">I</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E5:39_2"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">I1</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">I2</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E3:39_2"><i><font color="Green" title="E40">A11</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="E6:39_2"/><b>then </b>
( <font color="Maroon" title="c15">x</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="c10">I1</font> &amp; <font color="Maroon" title="c15">x</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="c12">I2</font> )
 <b>by </b><i><a class="ref" href="vectsp_7.html#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></i>;<br/><a NAME="E7:39_2"/><b>then </b>
( <font color="Maroon" title="c15">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">W1</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U4</a> of <font color="Maroon" title="c6">W1</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c6">W1</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c6">W1</font> #) &amp; <font color="Maroon" title="c15">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W2</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U4</a> of <font color="Maroon" title="c7">W2</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c7">W2</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c7">W2</font> #) )
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/><a NAME="E8:39_2"/><b>then </b><i><font color="Green" title="E42">A13</font></i>: 
( <font color="Maroon" title="c15">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="c6">W1</font> &amp; <font color="Maroon" title="c15">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="c7">W2</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/><a NAME="E9:39_2"/><b>then </b>
<font color="Maroon" title="c15">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="c6">W1</font> <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 <font color="Maroon" title="c7">W2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:39_2"/><b>then </b>
<font color="Maroon" title="c15">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"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</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/><a NAME="E11:39_2"/><b>then </b><i><font color="Green" title="E43">A14</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font></span>)</span>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U4</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font></span>)</span>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font></span>)</span>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E12:39_2"/><i><font color="Green" title="E44">A15</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">W1</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="c5">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="c16">x'</font> = <font color="Maroon" title="c15">x</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c5">V</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E8:39_2"><i><font color="Green" title="E42">A13</font></i></a></i>;<br/><a NAME="E14:39_2"/>
( <font color="Maroon" title="c8">I</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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font></span>)</span> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</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="c5">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="c17">I'</font> = <font color="Maroon" title="c8">I</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">V</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_2_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">I</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E2:39_2_1"/><b>then </b><i><font color="Green" title="E45">A16</font></i>: 
( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">I</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:39_2_1"/>
( <font color="Maroon" title="c8">I</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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font></span>)</span> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</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="c5">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/><a NAME="E4:39_2_1"/><b>then </b>
<font color="Maroon" title="c8">I</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="c5">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:39_2_1"/><b>then </b>
( <font color="Maroon" title="c18">y</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">V</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E2:39_2_1"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:39_2_1"/>
<font color="Maroon" title="c18">y</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">V</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E8:39_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="vectsp_9.html#E12:39_2"><i><font color="Green" title="E44">A15</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c18">Ix</font> = <font color="Maroon" title="c8">I</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E18:39_2"/>
( <font color="Maroon" title="c10">I1</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp; <font color="Maroon" title="c10">I1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">W1</font> )
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">I1'</font> = <font color="Maroon" title="c10">I1</font> as  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">V</font> <b>by </b><i><a class="ref" href="vectsp_9.html#T15" target="_self" title="VECTSP_9:th.15">Th15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_2_2"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">I</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E2:39_2_2"/><b>then </b>
( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">I</font> or <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:39_2_2"/><b>then </b>
( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">I1</font> or <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E9:39"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:39_2_2"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">I1'</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E3:39_2"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E21:39_2"/><b>then </b><i><font color="Green" title="E45">A17</font></i>: 
<font color="Maroon" title="c18">Ix</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">I1'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E22:39_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E23:39_2"/><b>then </b><i><font color="Green" title="E46">A18</font></i>: 
<font color="Maroon" title="c16">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Ix</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><font color="Maroon" title="c18">Ix</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<font color="Maroon" title="c8">I</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></i>
<br/>.= 
<font color="Maroon" title="c8">I</font>
<b>by </b><i><a class="txt" href="vectsp_9.html#E4:39_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>
;<br/><a NAME="E25:39_2"/><b>then </b><i><font color="Green" title="E47">A19</font></i>: 
not <font color="Maroon" title="c16">x'</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="c17">I'</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E21:39_2"><i><font color="Green" title="E45">A17</font></i></a>, <a class="txt" href="vectsp_9.html#E23:39_2"><i><font color="Green" title="E46">A18</font></i></a>, <a class="ref" href="vectsp_9.html#T18" target="_self" title="VECTSP_9:th.18">Th18</a>, <a class="ref" href="vectsp_7.html#T2" title="VECTSP_7:th.2">VECTSP_7:2</a></i>;<br/><a NAME="E26:39_2"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c8">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">I'</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T21" target="_self" title="VECTSP_9:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E27:39_2"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp_9.html#E5:39"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="vectsp_9.html#E11:39_2"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="vectsp_9.html#E25:39_2"><i><font color="Green" title="E47">A19</font></i></a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:39"/><b>then </b>
<font color="Maroon" title="c8">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">I1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">I2</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E19:39"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E22:39"/><b>then </b><i><font color="Green" title="E40">A20</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">I1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">I2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T64" title="CARD_2:th.64">CARD_2:64</a></i>;<br/>
<a NAME="E23:39"/>
( <font color="Maroon" title="c14">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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</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="c5">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="c15">A'</font> = <font color="Maroon" title="c14">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">V</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E25:39"/><i><font color="Green" title="E41">A21</font></i>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c15">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c14">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T21" target="_self" title="VECTSP_9:th.21">Th21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_3"/>
<font color="Maroon" title="c16">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"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span>
 ;<br/><a NAME="E2:39_3"/><b>then </b>
<font color="Maroon" title="c16">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</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="c17">w1</font>, <font color="Maroon" title="c18">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c5">V</font><b> such that </b><br/><a NAME="E4:39_3"/><i><font color="Green" title="E42">A22</font></i>: 
<font color="Maroon" title="c17">w1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font>
 <b>and </b><br/><a NAME="E5:39_3"/><i><font color="Green" title="E43">A23</font></i>: 
<font color="Maroon" title="c18">w2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W2</font>
 <b>and </b><br/><a NAME="E6:39_3"/><i><font color="Green" title="E44">A24</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">w1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">w2</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5" title="VECTSP_5:th.5">VECTSP_5:5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">w1</font> = <font color="Maroon" title="c17">w1</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c6">W1</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E4:39_3"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">w2</font> = <font color="Maroon" title="c18">w2</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c7">W2</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E5:39_3"><i><font color="Green" title="E43">A23</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="E9:39_3"/>
<font color="Maroon" title="c19">w1</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="c10">I1</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T14" target="_self" title="VECTSP_9:th.14">Th14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">K1</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c10">I1</font><b> such that </b><br/><a NAME="E11:39_3"/><i><font color="Green" title="E45">A25</font></i>: 
<font color="Maroon" title="c19">w1</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="c21">K1</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12" title="VECTSP_7:th.12">VECTSP_7:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">L1</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c5">V</font><b> such that </b><br/><a NAME="E13:39_3"/><i><font color="Green" title="E46">A26</font></i>: 
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c22">L1</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="c21">K1</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c22">L1</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="c21">K1</font> )
 <b>by </b><i><a class="ref" href="vectsp_9.html#T12" target="_self" title="VECTSP_9:th.12">Th12</a></i>;<br/><a NAME="E14:39_3"/>
<font color="Maroon" title="c20">w2</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="c12">I2</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T14" target="_self" title="VECTSP_9:th.14">Th14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">K2</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c12">I2</font><b> such that </b><br/><a NAME="E16:39_3"/><i><font color="Green" title="E47">A27</font></i>: 
<font color="Maroon" title="c20">w2</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="c23">K2</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12" title="VECTSP_7:th.12">VECTSP_7:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c24">L2</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c5">V</font><b> such that </b><br/><a NAME="E18:39_3"/><i><font color="Green" title="E48">A28</font></i>: 
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c24">L2</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="c23">K2</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c24">L2</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="c23">K2</font> )
 <b>by </b><i><a class="ref" href="vectsp_9.html#T12" target="_self" title="VECTSP_9:th.12">Th12</a></i>;<br/><b>set </b><font color="Maroon" title="c25">L</font> = <font color="Maroon" title="c22">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c24">L2</font>;<br/><a NAME="E19:39_3"/>
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c22">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">I1</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c24">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">I2</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E13:39_3"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="vectsp_9.html#E18:39_3"><i><font color="Green" title="E48">A28</font></i></a>, <a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/><a NAME="E20:39_3"/><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="c22">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c24">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="c22">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="c24">L2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c22">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="c24">L2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">I1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">I2</font> )
 <b>by </b><i><a class="ref" href="vectsp_6.html#T51" title="VECTSP_6:th.51">VECTSP_6:51</a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></i>;<br/><a NAME="E21:39_3"/><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="c22">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c24">L2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">I1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">I2</font>
 <b>by </b><i><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="c26">L</font> = <font color="Maroon" title="c22">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c24">L2</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c15">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="E23:39_3"/>
<font color="Maroon" title="c16">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="c26">L</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E6:39_3"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="vectsp_9.html#E11:39_3"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="vectsp_9.html#E13:39_3"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="vectsp_9.html#E16:39_3"><i><font color="Green" title="E47">A27</font></i></a>, <a class="txt" href="vectsp_9.html#E18:39_3"><i><font color="Green" title="E48">A28</font></i></a>, <a class="ref" href="vectsp_6.html#T77" title="VECTSP_6:th.77">VECTSP_6:77</a></i>;<br/><a NAME="E24:39_3"/><b>then </b>
<font color="Maroon" title="c16">x</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="c15">A'</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12" title="VECTSP_7:th.12">VECTSP_7:12</a></i>;<br/><b>hence </b><a NAME="E25:39_3"/>
<font color="Maroon" title="c16">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="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c15">A'</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:39"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</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 <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c15">A'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E28:39"/><b>then </b>
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c15">A'</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T35" title="VECTSP_4:th.35">VECTSP_4:35</a></i>;<br/>
<a NAME="E29:39"/><b>then </b><i><font color="Green" title="E42">A29</font></i>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c14">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E25:39"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="vectsp_4.html#T33" title="VECTSP_4:th.33">VECTSP_4:33</a></i>;<br/>
<a NAME="E30:39"/>
for <font color="Olive" title="b1">L</font> being   <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c14">A</font>  st  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span> holds <br/> <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b1">L</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="39_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">L</font> be    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c14">A</font>;<br/>

<b>assume </b><a NAME="E1:39_4"/><i><font color="Green" title="E43">A30</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c16">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span>
 ;<br/>

<a NAME="E2:39_4"/><i><font color="Green" title="E44">A31</font></i>: 
( <font color="Maroon" title="c6">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font> &amp; <font color="Maroon" title="c7">W2</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T11" title="VECTSP_5:th.11">VECTSP_5:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">W1'</font> = <font color="Maroon" title="c6">W1</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font> <b>by </b><i><a class="ref" href="vectsp_5.html#T11" title="VECTSP_5:th.11">VECTSP_5:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">W2'</font> = <font color="Maroon" title="c7">W2</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font> <b>by </b><i><a class="ref" href="vectsp_5.html#T11" title="VECTSP_5:th.11">VECTSP_5:11</a></i>;<br/>
<a NAME="E5:39_4"/><i><font color="Green" title="E45">A32</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">I1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">I2</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/>
<b>consider </b><font color="Maroon" title="c19">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span><b> such that </b><br/><a NAME="E7:39_4"/><i><font color="Green" title="E46">A33</font></i>: 
<font color="Maroon" title="c19">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E8:39_4"/><i><font color="Green" title="E47">A34</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">F</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="c16">L</font>
 <b>and </b><br/><a NAME="E9:39_4"/><i><font color="Green" title="E48">A35</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c16">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c19">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9" title="VECTSP_6:def.9">VECTSP_6:def 9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">B</font> = <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">I1</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">B</font> = <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">I1</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">F</font></span>)</span> <b>by </b><i><a class="txt" href="vectsp_9.html#E8:39_4"><i><font color="Green" title="E47">A34</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">P</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">F</font><b> such that </b><br/><a NAME="E12:39_4"/><i><font color="Green" title="E49">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">F</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">F</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c21">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">P</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E7:39_4"><i><font color="Green" title="E46">A33</font></i></a>, <a class="ref" href="finseq_3.html#T124" title="FINSEQ_3:th.124">FINSEQ_3:124</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">F1</font> = <font color="Maroon" title="c19">F</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <font color="Maroon" title="c24">F2</font> = <font color="Maroon" title="c19">F</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c21">B</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_3.html#T93" title="FINSEQ_3:th.93">FINSEQ_3:93</a></i>;<br/>
<a NAME="E14:39_4"/><i><font color="Green" title="E50">A37</font></i>: 
( <font color="Maroon" title="c23">F1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c24">F2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E7:39_4"><i><font color="Green" title="E46">A33</font></i></a>, <a class="ref" href="finseq_3.html#T94" title="FINSEQ_3:th.94">FINSEQ_3:94</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">L1</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font><b> such that </b><br/><a NAME="E16:39_4"/><i><font color="Green" title="E51">A38</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c25">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">F1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span>
 <b>and </b><br/><a NAME="E17:39_4"/><i><font color="Green" title="E52">A39</font></i>: 
<font color="Maroon" title="c25">L1</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c23">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c23">F1</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T8" target="_self" title="VECTSP_9:th.8">Th8</a></i>;<br/>
<a NAME="E18:39_4"/><i><font color="Green" title="E53">A40</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c23">F1</font></span>)</span> <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="c25">L1</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E14:39_4"><i><font color="Green" title="E50">A37</font></i></a>, <a class="txt" href="vectsp_9.html#E16:39_4"><i><font color="Green" title="E51">A38</font></i></a>, <a class="txt" href="vectsp_9.html#E17:39_4"><i><font color="Green" title="E52">A39</font></i></a>, <a class="ref" href="vectsp_9.html#T7" target="_self" title="VECTSP_9:th.7">Th7</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E19:39_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">F</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">X</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">F</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">F</font></span>)</span> ;<br/>
<font color="Maroon" title="c26">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> = 
<font color="Maroon" title="c26">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#T32" title="SUBSET_1:th.32">SUBSET_1:32</a></i>
<br/>.= 
<font color="Maroon" title="c21">B</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E22:39_4"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">B</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T72" title="FINSEQ_3:th.72">FINSEQ_3:72</a></i>;<br/>
<b>then </b><i><font color="Green" title="E54">A41</font></i>:  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c25">L1</font> = 
<font color="Maroon" title="c10">I1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E16:39_4"><i><font color="Green" title="E51">A38</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">I1</font>

;<br/>
<a NAME="E24:39_4"/><b>then </b><i><font color="Green" title="E55">A42</font></i>: 
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c25">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">I1</font> &amp; <font color="Maroon" title="c10">I1</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="c6">W1</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">K1</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c17">W1'</font><b> such that </b><br/><a NAME="E26:39_4"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c27">K1</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="c25">L1</font>
 <b>and </b><br/><a NAME="E27:39_4"/><i><font color="Green" title="E56">A43</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c27">K1</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="c25">L1</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E23:39_4"><i><font color="Green" title="E54">A41</font></i></a>, <a class="ref" href="vectsp_9.html#T13" target="_self" title="VECTSP_9:th.13">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">L2</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font><b> such that </b><br/><a NAME="E29:39_4"/><i><font color="Green" title="E57">A44</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c28">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">F2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span>
 <b>and </b><br/><a NAME="E30:39_4"/><i><font color="Green" title="E58">A45</font></i>: 
<font color="Maroon" title="c28">L2</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c24">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c24">F2</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T8" target="_self" title="VECTSP_9:th.8">Th8</a></i>;<br/>
<a NAME="E31:39_4"/><i><font color="Green" title="E59">A46</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c24">F2</font></span>)</span> <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="c28">L2</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E14:39_4"><i><font color="Green" title="E50">A37</font></i></a>, <a class="txt" href="vectsp_9.html#E29:39_4"><i><font color="Green" title="E57">A44</font></i></a>, <a class="txt" href="vectsp_9.html#E30:39_4"><i><font color="Green" title="E58">A45</font></i></a>, <a class="ref" href="vectsp_9.html#T7" target="_self" title="VECTSP_9:th.7">Th7</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">F2</font> = 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">I1</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E8:39_4"><i><font color="Green" title="E47">A34</font></i></a>, <a class="ref" href="finseq_3.html#T72" title="FINSEQ_3:th.72">FINSEQ_3:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">I1</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T47" title="XBOOLE_1:th.47">XBOOLE_1:47</a></i>
;<br/>
<a NAME="E33:39_4"/><b>then </b><i><font color="Green" title="E60">A47</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c28">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">I1</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E29:39_4"><i><font color="Green" title="E57">A44</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E34:39_4"/><b>then </b>
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c28">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">I2</font> &amp; <font color="Maroon" title="c12">I2</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="c7">W2</font> )
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E5:39_4"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="xboole_1.html#T43" title="XBOOLE_1:th.43">XBOOLE_1:43</a></i>;<br/>
<a NAME="E35:39_4"/><b>then </b>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c28">L2</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="c18">W2'</font>
 
<b>by </b><i><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="c29">K2</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c18">W2'</font><b> such that </b><br/><a NAME="E37:39_4"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c29">K2</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="c28">L2</font>
 <b>and </b><br/><a NAME="E38:39_4"/><i><font color="Green" title="E61">A48</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c29">K2</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="c28">L2</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T13" target="_self" title="VECTSP_9:th.13">Th13</a></i>;<br/>
<i><font color="Green" title="E62">A49</font></i>:  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span> = 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">F2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E1:39_4"><i><font color="Green" title="E43">A30</font></i></a>, <a class="txt" href="vectsp_9.html#E9:39_4"><i><font color="Green" title="E48">A35</font></i></a>, <a class="txt" href="vectsp_9.html#E12:39_4"><i><font color="Green" title="E49">A36</font></i></a>, <a class="ref" href="vectsp_9.html#T5" target="_self" title="VECTSP_9:th.5">Th5</a></i>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c23">F1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c24">F2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T37" title="VECTSP_6:th.37">VECTSP_6:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c25">L1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c28">L2</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E18:39_4"><i><font color="Green" title="E53">A40</font></i></a>, <a class="txt" href="vectsp_9.html#E31:39_4"><i><font color="Green" title="E59">A46</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
;<br/>
<b>then </b> <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c25">L1</font> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0: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="c28">L2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T63" title="VECTSP_1:th.63">VECTSP_1:63</a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0: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="c29">K2</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E38:39_4"><i><font color="Green" title="E61">A48</font></i></a>, <a class="ref" href="vectsp_4.html#T23" title="VECTSP_4:th.23">VECTSP_4:23</a></i>
;<br/>
<a NAME="E41:39_4"/><b>then </b>
(  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c27">K1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W2</font> &amp;  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c27">K1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> )
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E27:39_4"><i><font color="Green" title="E56">A43</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="E42:39_4"/><b>then </b>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c27">K1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></i>;<br/>
<a NAME="E43:39_4"/><b>then </b>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c27">K1</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="c8">I</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E5:39"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c30">KI</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c8">I</font><b> such that </b><br/><a NAME="E45:39_4"/><i><font color="Green" title="E63">A50</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c27">K1</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="c30">KI</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="E46:39_4"/>
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T29" title="VECTSP_5:th.29">VECTSP_5:29</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c31">LI</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font><b> such that </b><br/><a NAME="E48:39_4"/><i><font color="Green" title="E64">A51</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c31">LI</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="c30">KI</font>
 <b>and </b><br/><a NAME="E49:39_4"/><i><font color="Green" title="E65">A52</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c31">LI</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="c30">KI</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T12" target="_self" title="VECTSP_9:th.12">Th12</a></i>;<br/>
<a NAME="E50:39_4"/><i><font color="Green" title="E66">A53</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">LI</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c28">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="c31">LI</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="c28">L2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T51" title="VECTSP_6:th.51">VECTSP_6:51</a></i>;<br/>
<a NAME="E51:39_4"/><i><font color="Green" title="E67">A54</font></i>: 
<font color="Maroon" title="c8">I</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E13:39"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E52:39_4"/>
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c31">LI</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">I</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c28">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">I2</font> )
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E5:39_4"><i><font color="Green" title="E45">A32</font></i></a>, <a class="txt" href="vectsp_9.html#E33:39_4"><i><font color="Green" title="E60">A47</font></i></a>, <a class="txt" href="vectsp_9.html#E48:39_4"><i><font color="Green" title="E64">A51</font></i></a>, <a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a>, <a class="ref" href="xboole_1.html#T43" title="XBOOLE_1:th.43">XBOOLE_1:43</a></i>;<br/>
<a NAME="E53:39_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c31">LI</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="c28">L2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">I2</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E51:39_4"><i><font color="Green" title="E67">A54</font></i></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E54:39_4"/><b>then </b><i><font color="Green" title="E68">A55</font></i>: 
(  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">LI</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c28">L2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">I2</font> &amp; <font color="Maroon" title="c12">I2</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="c7">W2</font> )
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E50:39_4"><i><font color="Green" title="E66">A53</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E55:39_4"/><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="c31">LI</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c28">L2</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="c7">W2</font>
 
<b>by </b><i><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="c32">K</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c7">W2</font><b> such that </b><br/><a NAME="E57:39_4"/><i><font color="Green" title="E69">A56</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c32">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> <span class="p1">(<span class="default"><font color="Maroon" title="c31">LI</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c28">L2</font></span>)</span>
 <b>and </b><br/><a NAME="E58:39_4"/><i><font color="Green" title="E70">A57</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c32">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> <span class="p1">(<span class="default"><font color="Maroon" title="c31">LI</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c28">L2</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E2:39_4"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="vectsp_9.html#T13" target="_self" title="VECTSP_9:th.13">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">K</font> = <font color="Maroon" title="c32">K</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c12">I2</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E54:39_4"><i><font color="Green" title="E68">A55</font></i></a>, <a class="txt" href="vectsp_9.html#E57:39_4"><i><font color="Green" title="E69">A56</font></i></a>, <a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E60:39_4"/>
( <font color="Maroon" title="c10">I1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c10">I1</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> )
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c34">I1'</font> = <font color="Maroon" title="c10">I1</font> as  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span> <b>by </b><i><a class="txt" href="vectsp_9.html#E2:39_4"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="vectsp_9.html#T15" target="_self" title="VECTSP_9:th.15">Th15</a></i>;<br/>
<a NAME="E62:39_4"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c31">LI</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">I</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E48:39_4"><i><font color="Green" title="E64">A51</font></i></a>, <a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E63:39_4"/><b>then </b>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c31">LI</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c34">I1'</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E9:39"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E64:39_4"/><b>then </b><i><font color="Green" title="E71">A58</font></i>: 
<font color="Maroon" title="c31">LI</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c25">L1</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E24:39_4"><i><font color="Green" title="E55">A42</font></i></a>, <a class="txt" href="vectsp_9.html#E27:39_4"><i><font color="Green" title="E56">A43</font></i></a>, <a class="txt" href="vectsp_9.html#E45:39_4"><i><font color="Green" title="E63">A50</font></i></a>, <a class="txt" href="vectsp_9.html#E49:39_4"><i><font color="Green" title="E65">A52</font></i></a>, <a class="ref" href="matrlin.html#T9" title="MATRLIN:th.9">MATRLIN:9</a></i>;<br/>
<a NAME="E65:39_4"/><i><font color="Green" title="E72">A59</font></i>: 
<font color="Maroon" title="c12">I2</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c7">W2</font> = 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c31">LI</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c28">L2</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E27:39_4"><i><font color="Green" title="E56">A43</font></i></a>, <a class="txt" href="vectsp_9.html#E39:39_4"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="vectsp_9.html#E45:39_4"><i><font color="Green" title="E63">A50</font></i></a>, <a class="txt" href="vectsp_9.html#E49:39_4"><i><font color="Green" title="E65">A52</font></i></a>, <a class="ref" href="vectsp_4.html#T20" title="VECTSP_4:th.20">VECTSP_4:20</a></i>
<br/>.= 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c33">K</font>
<b>by </b><i><a class="txt" href="vectsp_9.html#E58:39_4"><i><font color="Green" title="E70">A57</font></i></a>, <a class="ref" href="vectsp_6.html#T77" title="VECTSP_6:th.77">VECTSP_6:77</a></i>
;<br/>
<a NAME="E67:39_4"/><b>then </b><i><font color="Green" title="E73">A60</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c28">L2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E57:39_4"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="vectsp_9.html#E64:39_4"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="vectsp_9.html#E65:39_4"><i><font color="Green" title="E72">A59</font></i></a>, <a class="ref" href="vectsp_7.html#D1" title="VECTSP_7:def.1">VECTSP_7:def 1</a></i>;<br/>
<a NAME="E68:39_4"/><i><font color="Green" title="E74">A61</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c25">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="c28">L2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E23:39_4"><i><font color="Green" title="E54">A41</font></i></a>, <a class="txt" href="vectsp_9.html#E33:39_4"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></i>;<br/>
<a NAME="E69:39_4"/><i><font color="Green" title="E75">A62</font></i>: 
<font color="Maroon" title="c10">I1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">I1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c25">L1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c28">L2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">I1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">I1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E23:39_4"><i><font color="Green" title="E54">A41</font></i></a>, <a class="txt" href="vectsp_9.html#E33:39_4"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="vectsp_9.html#E69:39_4"><i><font color="Green" title="E75">A62</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/>
<a NAME="E71:39_4"/><b>then </b><i><font color="Green" title="E76">A63</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c25">L1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c28">L2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_4_8"/>
not  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c28">L2</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c35">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_4_8"/><i><font color="Green" title="E77">A64</font></i>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">L</font>
 <b>and </b><br/><a NAME="E4:39_4_8"/><i><font color="Green" title="E78">A65</font></i>: 
not <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c28">L2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c36">x</font> = <font color="Maroon" title="c35">x</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span> <b>by </b><i><a class="txt" href="vectsp_9.html#E3:39_4_8"><i><font color="Green" title="E77">A64</font></i></a></i>;<br/><i><font color="Green" title="E79">A66</font></i>:  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">GF</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c28">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">x</font>
<b>by </b><i><a class="txt" href="vectsp_9.html#E4:39_4_8"><i><font color="Green" title="E78">A65</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="c25">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#D11" title="VECTSP_6:def.11">VECTSP_6:def 11</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_4_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_4_8_2"/>
( <font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c25">L1</font> or <font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c28">L2</font> )
 </span><b>by </b><i><a class="txt" href="vectsp_9.html#E68:39_4"><i><font color="Green" title="E74">A61</font></i></a>, <a class="txt" href="vectsp_9.html#E3:39_4_8"><i><font color="Green" title="E77">A64</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_4_8_2_1"><b>suppose </b></a><a NAME="E1:39_4_8_2_1"/><i><font color="Green" title="E80">A67</font></i>: 
<font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c25">L1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c37">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span><b> such that </b><br/><a NAME="E3:39_4_8_2_1"/><i><font color="Green" title="E81">A68</font></i>: 
( <font color="Maroon" title="c36">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">v</font> &amp; <font color="Maroon" title="c25">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c37">v</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">GF</font> )
 <b>by </b><i><a class="ref" href="vectsp_6.html#T19" title="VECTSP_6:th.19">VECTSP_6:19</a></i>;<br/><a NAME="E4:39_4_8_2_1"/>
not <font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c28">L2</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E71:39_4"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="vectsp_9.html#E1:39_4_8_2_1"><i><font color="Green" title="E80">A67</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:39_4_8_2_1"/><b>then </b>
<font color="Maroon" title="c28">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">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="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="E6:39_4_8_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp_9.html#E6:39_4_8"><i><font color="Green" title="E79">A66</font></i></a>, <a class="txt" href="vectsp_9.html#E3:39_4_8_2_1"><i><font color="Green" title="E81">A68</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_4_8_2_2"><b>suppose </b></a><a NAME="E1:39_4_8_2_2"/><i><font color="Green" title="E80">A69</font></i>: 
<font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c28">L2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c37">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span><b> such that </b><br/><a NAME="E3:39_4_8_2_2"/><i><font color="Green" title="E81">A70</font></i>: 
( <font color="Maroon" title="c36">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">v</font> &amp; <font color="Maroon" title="c28">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c37">v</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">GF</font> )
 <b>by </b><i><a class="ref" href="vectsp_6.html#T19" title="VECTSP_6:th.19">VECTSP_6:19</a></i>;<br/><a NAME="E4:39_4_8_2_2"/>
not <font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c25">L1</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E71:39_4"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="vectsp_9.html#E1:39_4_8_2_2"><i><font color="Green" title="E80">A69</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:39_4_8_2_2"/><b>then </b>
<font color="Maroon" title="c25">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">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="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="E6:39_4_8_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp_9.html#E6:39_4_8"><i><font color="Green" title="E79">A66</font></i></a>, <a class="txt" href="vectsp_9.html#E3:39_4_8_2_2"><i><font color="Green" title="E81">A70</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E73:39_4"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">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="vectsp_9.html#E67:39_4"><i><font color="Green" title="E73">A60</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:39"/><b>then </b>
<font color="Maroon" title="c14">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D1" title="VECTSP_7:def.1">VECTSP_7:def 1</a></i>;<br/>
<a NAME="E32:39"/><b>then </b>
<font color="Maroon" title="c14">A</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E29:39"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c7">W2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">I1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">I2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">I</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">I</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E6:39"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="vectsp_9.html#E16:39"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="vectsp_9.html#E22:39"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="vectsp_9.html#D2" target="_self" title="VECTSP_9:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W2</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E15:39"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="vectsp_9.html#D2" target="_self" title="VECTSP_9:def.2">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E34:39"/>
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">W2</font></span>)</span>
 ;<br/>


</div>
