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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_7.html#K8" title="EUCLID_7:func.8">RN_Base</a> <font color="Maroon" title="c1">n</font> is    <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of  <a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c2">B</font> =  <a href="euclid_7.html#K8" title="EUCLID_7:func.8">RN_Base</a> <font color="Maroon" title="c1">n</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="real_ns1.html#D6" title="REAL_NS1:def.6">REAL_NS1:def 6</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">V</font> =  <a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E2:107"/><i><font color="Green" title="E58">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">B</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_3.html#D1" title="RUSUB_3:def.1">RUSUB_3:def 1</a></i>;<br/>
<div><i><font color="Green" title="E59">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:107_1"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c2">B</font></span>)</span>
 ; <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="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:107_1"/><i><font color="Green" title="E60">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c2">B</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">x0</font> = <font color="Maroon" title="c4">x</font> as    <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 <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:107_1"><i><font color="Green" title="E60">A4</font></i></a></i>;<br/><a NAME="E5:107_1"/>
 ex <font color="Olive" title="b1">l</font> being   <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">B</font> st <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">l</font>
 <b>by </b><i><a class="ref" href="euclid_7.html#T47" target="_self" title="EUCLID_7:th.47">Th53</a></i>;<br/><b>hence </b><a NAME="E6:107_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:107"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="#E3:107_1"><i><font color="Green" title="E60">A4</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>
<a NAME="E4:107"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c2">B</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:107_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c2">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/>

<a NAME="E2:107_2"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">B</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">l</font>
 
<b>by </b><i><a class="txt" href="#E2:107"><i><font color="Green" title="E58">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:107_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:107"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:107"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E6:107"/><b>then </b>
 <a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#T26" title="RUSUB_1:th.26">RUSUB_1:26</a></i>;<br/>
<b>hence </b><a NAME="E7:107"/>
 <a href="euclid_7.html#K8" title="EUCLID_7:func.8">RN_Base</a> <font color="Maroon" title="c1">n</font> is    <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of  <a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="rusub_3.html#D2" title="RUSUB_3:def.2">RUSUB_3:def 2</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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