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<b>let </b><font color="Maroon" title="c1">GF</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">GF</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b2">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> holds <br/> ex <font color="Olive" title="b3">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font> st <br/>( <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive" title="b1">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U5</a> of <font color="Olive" title="b1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Olive" title="b1">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Olive" title="b1">V</font> #) )</span><br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">GF</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> holds <br/> ex <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b2">B</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U5</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #) )</span><br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> implies  ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U5</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #) ) )</span><br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U5</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #) )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> );<br/>
<b>consider </b><font color="Maroon" title="c4">Q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:21"/><i><font color="Green" title="E12">A2</font></i>: 
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> iff ( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Z</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:21"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c4">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c5">Z</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> implies  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:21_1"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c5">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:21_1"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Q</font>
 <b>and </b><br/><a NAME="E3:21_1"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c5">Z</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font></span><br/><b>set </b><font color="Maroon" title="c6">W</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Z</font>;<br/><a NAME="E4:21_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Z</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> )</span><br/>

<b>assume </b><a NAME="E1:21_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#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="c2">V</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:21_1_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <b>and </b><br/><a NAME="E4:21_1_1"/><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:21_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E2:21_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="#E4:21_1_1"><i><font color="Green" title="E18">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:21_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="#E3:21_1_1"><i><font color="Green" title="E17">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c7">W</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> ;<br/><b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Z</font>;<br/><a NAME="E7:21_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <b>by </b><i><a class="txt" href="#E1:21_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="#E2:21_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E8:21_1"/><b>then </b><i><font color="Green" title="E17">A9</font></i>: 
 ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> )
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/><a NAME="E9:21_1"/>
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="#E1:21_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:21_1"/><b>then </b><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="#E8:21_1"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:21_1"/>
<font color="Maroon" title="c7">W</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">l</font> be    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c7">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_7.html#D1" target="_self" title="VECTSP_7:def.1">VECTSP_7:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c9">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> <font color="Maroon" title="c2">V</font> implies  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:21_1_2"/><i><font color="Green" title="E19">A11</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c9">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> <font color="Maroon" title="c2">V</font>
 <b>and </b><br/><a NAME="E2:21_1_2"/><i><font color="Green" title="E20">A12</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> : ( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c10">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:21_1_2"/><i><font color="Green" title="E21">A13</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">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="c9">l</font>
 <b>and </b><br/><a NAME="E5:21_1_2"/><i><font color="Green" title="E22">A14</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c9">l</font> holds <br/><font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c11">M</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">f</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="#E2:21_1_2"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="#E4:21_1_2"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">F</font> being    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of <font color="Maroon" title="c11">M</font>;<br/>
<div><i><font color="Green" title="E23">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_2_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:21_1_2_1"/><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">f</font>
 <b>and </b><br/><a NAME="E4:21_1_2_1"/><i><font color="Green" title="E25">A17</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</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="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:21_1_2_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W</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>then consider </b><font color="Maroon" title="c14">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:21_1_2_1"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">X</font>
 <b>and </b><br/><a NAME="E8:21_1_2_1"/><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 <b>by </b><i><a class="txt" href="#E4:21_1_2"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E3:21_1_2_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">X</font> = <font color="Maroon" title="c14">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E2:21_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="#E8:21_1_2_1"><i><font color="Green" title="E27">A19</font></i></a></i>;<br/><a NAME="E10:21_1_2_1"/>
<font color="Maroon" title="c15">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> : ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E7:21_1_2_1"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="#E8:21_1_2_1"><i><font color="Green" title="E27">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E11:21_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:21_1_2"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E5:21_1_2"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="#E3:21_1_2_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="#E4:21_1_2_1"><i><font color="Green" title="E25">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c13">S</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font>;<br/>
<a NAME="E9:21_1_2"/><i><font color="Green" title="E24">A20</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c11">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="#E8:21_1_2"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="rlvect_3.html#T35" title="RLVECT_3:th.35">RLVECT_3:35</a></i>;<br/>
<a NAME="E10:21_1_2"/><b>then </b><i><font color="Green" title="E25">A21</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<div><i><font color="Green" title="E26">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font> implies <font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> )</span><br/><b>assume </b><a NAME="E1:21_1_2_2"/>
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font></span><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="E3:21_1_2_2"/><i><font color="Green" title="E27">A23</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">F</font>
 <b>and </b><br/><a NAME="E4:21_1_2_2"/><i><font color="Green" title="E28">A24</font></i>: 
<font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">X</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:21_1_2_2"/><i><font color="Green" title="E29">A25</font></i>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">f</font>
 <b>and </b><br/><a NAME="E7:21_1_2_2"/><i><font color="Green" title="E30">A26</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">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="#E9:21_1_2"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="#E3:21_1_2_2"><i><font color="Green" title="E27">A23</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:21_1_2_2"/><i><font color="Green" title="E31">A27</font></i>: 
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="#E8:21_1_2"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="#E9:21_1_2"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="#E3:21_1_2_2"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="#E4:21_1_2_2"><i><font color="Green" title="E28">A24</font></i></a>, <a class="ref" href="orders_1.html#D1" title="ORDERS_1:def.1">ORDERS_1:def 1</a></i>;<br/><a NAME="E9:21_1_2_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> : ( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:21_1_2"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E5:21_1_2"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="#E6:21_1_2_2"><i><font color="Green" title="E29">A25</font></i></a>, <a class="txt" href="#E7:21_1_2_2"><i><font color="Green" title="E30">A26</font></i></a></i>;<br/><a NAME="E10:21_1_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">X</font> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> )
 <b>by </b><i><a class="txt" href="#E8:21_1_2_2"><i><font color="Green" title="E31">A27</font></i></a></i>;<br/><b>hence </b><a NAME="E11:21_1_2_2"/>
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E27">A28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">X</font>, <font color="Maroon" title="c15">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font> &amp; <font color="Maroon" title="c15">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font> &amp; not <font color="Maroon" title="c14">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Y</font> implies <font color="Maroon" title="c15">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">X</font> )</span><br/><b>assume </b><a NAME="E1:21_1_2_3"/>
( <font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font> &amp; <font color="Maroon" title="c15">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Y</font> or <font color="Maroon" title="c15">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">X</font> )</span><br/><a NAME="E2:21_1_2_3"/><b>then </b>
( <font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> &amp; <font color="Maroon" title="c15">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> )
 <b>by </b><i><a class="txt" href="#E11:21_1_2"><i><font color="Green" title="E26">A22</font></i></a></i>;<br/><a NAME="E3:21_1_2_3"/><b>then </b>
<font color="Maroon" title="c14">X</font>,<font color="Maroon" title="c15">Y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="txt" href="#E3:21_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></i>;<br/><b>hence </b><a NAME="E4:21_1_2_3"/>
( <font color="Maroon" title="c14">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Y</font> or <font color="Maroon" title="c15">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E13:21_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="#E4:21_1_2"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="#E9:21_1_2"><i><font color="Green" title="E24">A20</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/>
<a NAME="E14:21_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/>
<a NAME="E15:21_1_2"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font>
 
<b>by </b><i><a class="txt" href="#E10:21_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="#E12:21_1_2"><i><font color="Green" title="E27">A28</font></i></a>, <a class="ref" href="card_2.html#T81" title="CARD_2:th.81">CARD_2:81</a></i>;<br/>
<a NAME="E16:21_1_2"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 
<b>by </b><i><a class="txt" href="#E11:21_1_2"><i><font color="Green" title="E26">A22</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E18:21_1_2"/><i><font color="Green" title="E28">A29</font></i>: 
<font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font></span>)</span>
 <b>and </b><br/><a NAME="E19:21_1_2"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">B</font>
 <b>and </b><br/><a NAME="E20:21_1_2"/><i><font color="Green" title="E29">A30</font></i>: 
<font color="Maroon" title="c14">B</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E2:21_1"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/>
<a NAME="E21:21_1_2"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c15">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="c9">l</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:21_1_2_4"/><i><font color="Green" title="E30">A31</font></i>: 
<font color="Maroon" title="c15">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="c9">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font></span>)</span></span><br/>

<a NAME="E2:21_1_2_4"/><b>then </b><i><font color="Green" title="E31">A32</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 
<b>by </b><i><a class="txt" href="#E4:21_1_2"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">X</font> = <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>;<br/>
<a NAME="E3:21_1_2_4"/><i><font color="Green" title="E32">A33</font></i>: 
<font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 
<b>by </b><i><a class="txt" href="#E8:21_1_2"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="#E2:21_1_2_4"><i><font color="Green" title="E31">A32</font></i></a>, <a class="ref" href="orders_1.html#D1" title="ORDERS_1:def.1">ORDERS_1:def 1</a></i>;<br/>
<a NAME="E4:21_1_2_4"/>
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> : ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E5:21_1_2"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="#E1:21_1_2_4"><i><font color="Green" title="E30">A31</font></i></a></i>;<br/>
<a NAME="E5:21_1_2_4"/><b>then </b><i><font color="Green" title="E33">A34</font></i>: 
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> )
 
<b>by </b><i><a class="txt" href="#E3:21_1_2_4"><i><font color="Green" title="E32">A33</font></i></a></i>;<br/>
<a NAME="E6:21_1_2_4"/>
<font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font>
 
<b>by </b><i><a class="txt" href="#E9:21_1_2"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="#E2:21_1_2_4"><i><font color="Green" title="E31">A32</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:21_1_2_4"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:21_1_2_4"><i><font color="Green" title="E33">A34</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E22:21_1_2"/><b>then </b>
<font color="Maroon" title="c9">l</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c14">B</font>
 
<b>by </b><i><a class="txt" href="#E18:21_1_2"><i><font color="Green" title="E28">A29</font></i></a>, <a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/>
<b>hence </b><a NAME="E23:21_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:21_1_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="#E2:21_1_2"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="#E20:21_1_2"><i><font color="Green" title="E29">A30</font></i></a>, <a class="ref" href="vectsp_7.html#D1" target="_self" title="VECTSP_7:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:21_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E10:21_1"><i><font color="Green" title="E18">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c5">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:21"/><i><font color="Green" title="E14">A35</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <b>and </b><br/><a NAME="E8:21"/><i><font color="Green" title="E15">A36</font></i>: 
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Olive" title="b1">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">X</font> holds <br/> not <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font>
 <b>by </b><i><a class="txt" href="#E4:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="orders_1.html#T177" title="ORDERS_1:th.177">ORDERS_1:177</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E10:21"/><i><font color="Green" title="E16">A37</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X</font>
 <b>and </b><br/><a NAME="E11:21"/><i><font color="Green" title="E17">A38</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font>
 <b>and </b><br/><a NAME="E12:21"/><i><font color="Green" title="E18">A39</font></i>: 
<font color="Maroon" title="c6">B</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E7:21"><i><font color="Green" title="E14">A35</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c6">B</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U5</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #) )</span><br/>

<b>thus </b><a NAME="E13:21"/>
( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c6">B</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> )
 <b>by </b><i><a class="txt" href="#E11:21"><i><font color="Green" title="E17">A38</font></i></a>, <a class="txt" href="#E12:21"><i><font color="Green" title="E18">A39</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U5</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #)</span><br/>

<b>assume </b><a NAME="E14:21"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U5</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c7">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E16:21"/><i><font color="Green" title="E19">A40</font></i>: 
( ( <font color="Maroon" title="c7">v</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="c6">B</font> &amp; not <font color="Maroon" title="c7">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font> ) or ( <font color="Maroon" title="c7">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font> &amp; not <font color="Maroon" title="c7">v</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="c6">B</font> ) )
 <b>by </b><i><a class="ref" href="vectsp_4.html#T38" title="VECTSP_4:th.38">VECTSP_4:38</a></i>;<br/>
<a NAME="E17:21"/><i><font color="Green" title="E20">A41</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">l</font> be    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_7.html#D1" target="_self" title="VECTSP_7:def.1">VECTSP_7:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c8">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> <font color="Maroon" title="c2">V</font> implies  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green" title="E21">A42</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c8">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> <font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_1_1"/>
( <font color="Maroon" title="c7">v</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="c8">l</font> or  not <font color="Maroon" title="c7">v</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="c8">l</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_1_1_1"/><i><font color="Green" title="E22">A43</font></i>: 
<font color="Maroon" title="c7">v</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="c8">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font> = <font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/><b>consider </b><font color="Maroon" title="c9">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font><b> such that </b><br/><a NAME="E3:21_2_1_1_1"/><i><font color="Green" title="E23">A44</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</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>and </b><br/><a NAME="E4:21_2_1_1_1"/><i><font color="Green" title="E24">A45</font></i>: 
for <font color="Olive" title="b1">u</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v</font> holds <br/><font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">u</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S6" title="FUNCT_2:sch.6">FUNCT_2:sch 6</a>();<br/></i><b>reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">u</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</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> )</span><br/><b>assume </b><a NAME="E1:21_2_1_1_1_1"/>
not <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</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></span><br/><a NAME="E2:21_2_1_1_1_1"/><b>then </b>
(  not <font color="Maroon" title="c11">u</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="c8">l</font> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E3:21_2_1_1_1_1"/><b>then </b>
( ( <font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</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> &amp; <font color="Maroon" title="c11">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v</font> ) or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:21_2_1_1_1_1"/>
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">GF</font>
 <b>by </b><i><a class="txt" href="#E3:21_2_1_1_1"><i><font color="Green" title="E23">A44</font></i></a>, <a class="txt" href="#E4:21_2_1_1_1"><i><font color="Green" title="E24">A45</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">f</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_6.html#D4" title="VECTSP_6:def.4">VECTSP_6:def 4</a></i>;<br/><a NAME="E8:21_2_1_1_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c11">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">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="c11">f</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )</span><br/>

<b>assume </b><a NAME="E1:21_2_1_1_1_2"/>
<font color="Maroon" title="c12">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="c11">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font></span><br/>

<b>then consider </b><font color="Maroon" title="c13">u</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:21_2_1_1_1_2"/><i><font color="Green" title="E25">A46</font></i>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font>
 <b>and </b><br/><a NAME="E4:21_2_1_1_1_2"/><i><font color="Green" title="E26">A47</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">u</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>
 ;<br/>
<a NAME="E5:21_2_1_1_1_2"/>
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">u</font>
 
<b>by </b><i><a class="txt" href="#E3:21_2_1_1_1"><i><font color="Green" title="E23">A44</font></i></a>, <a class="txt" href="#E4:21_2_1_1_1"><i><font color="Green" title="E24">A45</font></i></a>, <a class="txt" href="#E4:21_2_1_1_1_2"><i><font color="Green" title="E26">A47</font></i></a></i>;<br/>
<a NAME="E6:21_2_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c13">u</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="c8">l</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="#E4:21_2_1_1_1_2"><i><font color="Green" title="E26">A47</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="E7:21_2_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> or <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:21_2_1_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="#E3:21_2_1_1_1"><i><font color="Green" title="E23">A44</font></i></a>, <a class="txt" href="#E3:21_2_1_1_1_2"><i><font color="Green" title="E25">A46</font></i></a>, <a class="txt" href="#E4:21_2_1_1_1_2"><i><font color="Green" title="E26">A47</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c11">f</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c6">B</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>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">GF</font>;<br/><b>consider </b><font color="Maroon" title="c13">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font><b> such that </b><br/><a NAME="E11:21_2_1_1_1"/><i><font color="Green" title="E25">A48</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span>
 <b>and </b><br/><a NAME="E12:21_2_1_1_1"/><i><font color="Green" title="E26">A49</font></i>: 
for <font color="Olive" title="b1">u</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v</font> holds <br/><font color="Maroon" title="c13">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">u</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S6" title="FUNCT_2:sch.6">FUNCT_2:sch 6</a>();<br/></i><b>reconsider </b><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">g</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">u</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">u</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> )</span><br/><b>assume </b><a NAME="E1:21_2_1_1_1_3"/>
not <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">u</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></span><br/><a NAME="E2:21_2_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:21_2_1_1_1_3"/>
<font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">GF</font>
 <b>by </b><i><a class="txt" href="#E12:21_2_1_1_1"><i><font color="Green" title="E26">A49</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c14">g</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_6.html#D4" title="VECTSP_6:def.4">VECTSP_6:def 4</a></i>;<br/><a NAME="E16:21_2_1_1_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c15">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_4"><b>proof </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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c16">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="c15">g</font> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:21_2_1_1_1_4"/>
<font color="Maroon" title="c16">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="c15">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<a NAME="E2:21_2_1_1_1_4"/><b>then </b>
 ex <font color="Olive" title="b1">u</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c15">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</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> )
 
;<br/>
<a NAME="E3:21_2_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 
<b>by </b><i><a class="txt" href="#E12:21_2_1_1_1"><i><font color="Green" title="E26">A49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:21_2_1_1_1_4"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c16">g</font> = <font color="Maroon" title="c15">g</font> as    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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="E18:21_2_1_1_1"/><i><font color="Green" title="E27">A50</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">g</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c8">l</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">u</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_6.html#D10" title="VECTSP_6:def.10">VECTSP_6:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_1_1_1_5_1_1"/>
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> or <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">u</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_1_1_1_5_1_1_1"/><i><font color="Green" title="E28">A51</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span><br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T73" title="VECTSP_6:th.73">VECTSP_6:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">GF</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:21_2_1_1_1"><i><font color="Green" title="E23">A44</font></i></a>, <a class="txt" href="#E11:21_2_1_1_1"><i><font color="Green" title="E25">A48</font></i></a>, <a class="txt" href="#E1:21_2_1_1_1_5_1_1_1"><i><font color="Green" title="E28">A51</font></i></a>, <a class="ref" href="rlvect_1.html#D12" title="RLVECT_1:def.12">RLVECT_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">GF</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T30" title="RLVECT_1:th.30">RLVECT_1:30</a></i>
<br/>.= 
<font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font>
<b>by </b><i><a class="txt" href="#E1:21_2_1_1_1_5_1_1_1"><i><font color="Green" title="E28">A51</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_1_1_1_5_1_1_2"/><i><font color="Green" title="E28">A52</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span><br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T73" title="VECTSP_6:th.73">VECTSP_6:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:21_2_1_1_1"><i><font color="Green" title="E24">A45</font></i></a>, <a class="txt" href="#E1:21_2_1_1_1_5_1_1_2"><i><font color="Green" title="E28">A52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">GF</font></span>)</span>
<b>by </b><i><a class="txt" href="#E12:21_2_1_1_1"><i><font color="Green" title="E26">A49</font></i></a>, <a class="txt" href="#E1:21_2_1_1_1_5_1_1_2"><i><font color="Green" title="E28">A52</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_2_1_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E19:21_2_1_1_1"/><i><font color="Green" title="E28">A53</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="txt" href="#E11:21_2_1_1_1"><i><font color="Green" title="E25">A48</font></i></a>, <a class="ref" href="vectsp_6.html#T43" title="VECTSP_6:th.43">VECTSP_6:43</a></i>;<br/><a NAME="E20:21_2_1_1_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c16">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:21_2"><i><font color="Green" title="E21">A42</font></i></a>, <a class="txt" href="#E18:21_2_1_1_1"><i><font color="Green" title="E27">A50</font></i></a>, <a class="ref" href="vectsp_6.html#T80" title="VECTSP_6:th.80">VECTSP_6:80</a></i>;<br/><b>then </b> <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c12">f</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span> <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="c16">g</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_2.html#T22" title="VECTSP_2:th.22">VECTSP_2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font>
<b>by </b><i><a class="txt" href="#E19:21_2_1_1_1"><i><font color="Green" title="E28">A53</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/><a NAME="E22:21_2_1_1_1"/><b>then </b><i><font color="Green" title="E29">A54</font></i>: 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</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="c6">B</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12" target="_self" title="VECTSP_7:th.12">Th12</a></i>;<br/><a NAME="E23:21_2_1_1_1"/>
<font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">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="txt" href="#E1:21_2_1_1_1"><i><font color="Green" title="E22">A43</font></i></a>, <a class="ref" href="vectsp_6.html#T20" title="VECTSP_6:th.20">VECTSP_6:20</a></i>;<br/><a NAME="E24:21_2_1_1_1"/><b>then </b><i><font color="Green" title="E30">A55</font></i>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <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_2.html#T34" title="VECTSP_2:th.34">VECTSP_2:34</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">GF</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">v</font>
<b>by </b><i><a class="txt" href="#E24:21_2_1_1_1"><i><font color="Green" title="E30">A55</font></i></a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/><b>hence </b><a NAME="E26:21_2_1_1_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">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="#E16:21"><i><font color="Green" title="E19">A40</font></i></a>, <a class="txt" href="#E22:21_2_1_1_1"><i><font color="Green" title="E29">A54</font></i></a>, <a class="ref" href="rlvect_1.html#T3" title="RLVECT_1:th.3">RLVECT_1:3</a>, <a class="ref" href="vectsp_4.html#T29" title="VECTSP_4:th.29">VECTSP_4:29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_1_1_2"/><i><font color="Green" title="E22">A56</font></i>: 
not <font color="Maroon" title="c7">v</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="c8">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:21_2_1_1_2"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">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="c8">l</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )</span><br/>

<b>assume </b><a NAME="E1:21_2_1_1_2_1"/><i><font color="Green" title="E23">A57</font></i>: 
<font color="Maroon" title="c9">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="c8">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font></span><br/>

<a NAME="E2:21_2_1_1_2_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<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="E3:21_2_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="#E1:21_2_1_1_2_1"><i><font color="Green" title="E23">A57</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:21_2_1_1_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="#E1:21_2_1_1_2"><i><font color="Green" title="E22">A56</font></i></a>, <a class="txt" href="#E1:21_2_1_1_2_1"><i><font color="Green" title="E23">A57</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:21_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">l</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c6">B</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>hence </b><a NAME="E4:21_2_1_1_2"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">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="#E12:21"><i><font color="Green" title="E18">A39</font></i></a>, <a class="txt" href="#E1:21_2"><i><font color="Green" title="E21">A42</font></i></a>, <a class="ref" href="vectsp_7.html#D1" target="_self" title="VECTSP_7:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_2"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E18:21"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E19:21"/><b>then </b><i><font color="Green" title="E21">A58</font></i>: 
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="#E16:21"><i><font color="Green" title="E19">A40</font></i></a>, <a class="ref" href="vectsp_7.html#T13" target="_self" title="VECTSP_7:th.13">Th13</a>, <a class="ref" href="rlvect_1.html#T3" title="RLVECT_1:th.3">RLVECT_1:3</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E20:21"/>
<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E21:21"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="#E11:21"><i><font color="Green" title="E17">A38</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E22:21"/><b>then </b>
<font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 
<b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E17:21"><i><font color="Green" title="E20">A41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E23:21"/>
contradiction
 <b>by </b><i><a class="txt" href="#E8:21"><i><font color="Green" title="E15">A36</font></i></a>, <a class="txt" href="#E10:21"><i><font color="Green" title="E16">A37</font></i></a>, <a class="txt" href="#E19:21"><i><font color="Green" title="E21">A58</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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