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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <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> <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">x1</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font> <a href="euclidlp.html#NR4" title="EUCLIDLP:NR.4">are_ldependent2</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> be    <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> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x1</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c3">x2</font> implies <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="euclidlp.html#NR4" title="EUCLIDLP:NR.4">are_ldependent2</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:50"/>
<font color="Maroon" title="c2">x1</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c3">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="euclidlp.html#NR4" title="EUCLIDLP:NR.4">are_ldependent2</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:50"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="euclidlp.html#R3" title="EUCLIDLP:pred.3">are_lindependent2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> = 
<font color="Maroon" title="c2">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="euclid_4.html#T3" title="EUCLID_4:th.3">EUCLID_4:3</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T8" target="_self" title="EUCLIDLP:th.8">Th8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:50"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="euclidlp.html#D2" target="_self" title="EUCLIDLP:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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