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<b>let </b><font color="Maroon" title="c1">B0</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c1">B0</font> is <a href="euclid_7.html#V5" title="EUCLID_7:attr.5">orthogonal_basis</a> implies <font color="Maroon" title="c1">B0</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="E1:38"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">B0</font> is <a href="euclid_7.html#V5" title="EUCLID_7:attr.5">orthogonal_basis</a>
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c1">B0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">B0</font>;<br/>
<a NAME="E4:38"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font>
 
<b>by </b><i><a class="txt" href="#E2:38"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">x0</font> = <font color="Maroon" title="c2">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ;<br/>
<a NAME="E6:38"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#T2" title="EUCLID:th.2">EUCLID:2</a></i>;<br/>
<a NAME="E7:38"/><b>then </b><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
;<br/>
<a NAME="E8:38"/><i><font color="Green" title="E26">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T2" title="EUCLID:th.2">EUCLID:2</a></i>;<br/>
<a NAME="E9:38"/><b>then </b>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="euclid_2.html#T17" title="EUCLID_2:th.17">EUCLID_2:17</a></i>;<br/>
<a NAME="E10:38"/><b>then </b><i><font color="Green" title="E27">A5</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="euclid_2.html#T16" title="EUCLID_2:th.16">EUCLID_2:16</a></i>;<br/>
<a NAME="E11:38"/>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E8:38"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="euclid.html#T2" title="EUCLID:th.2">EUCLID:2</a></i>;<br/>
<b>hence </b><a NAME="E12:38"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E2:38"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E7:38"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="#E10:38"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="euclid_7.html#D4" target="_self" title="EUCLID_7:def.4">Def5</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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