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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y0</font>, <font color="Olive" title="b3">y1</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">x</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Olive" title="b2">y0</font> &amp; <font color="Olive" title="b2">y0</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Olive" title="b3">y1</font> holds <br/><font color="Olive" title="b1">x</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Olive" title="b3">y1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y0</font>, <font color="Maroon" title="c4">y1</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c3">y0</font> &amp; <font color="Maroon" title="c3">y0</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c4">y1</font> implies <font color="Maroon" title="c2">x</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c4">y1</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c3">y0</font>
 <span class="kw">and </span><br/><a NAME="E2:49"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c3">y0</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c4">y1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c4">y1</font></span><br/>

<a NAME="E3:49"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y0</font> <a href="rvsum_1.html#R1" title="RVSUM_1:pred.1">are_orthogonal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">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="E5:49"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></span>;<br/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y1</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> = 
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y0</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="euclid_4.html#T22" title="EUCLID_4:th.22">EUCLID_4:22</a></span>
<br/>.= 
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="rvsum_1.html#D17" title="RVSUM_1:def.17">RVSUM_1:def 17</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<a NAME="E7:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y1</font> <a href="rvsum_1.html#R1" title="RVSUM_1:pred.1">are_orthogonal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D17" title="RVSUM_1:def.17">RVSUM_1:def 17</a></span>;<br/>
<a NAME="E8:49"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a>, <a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:49"/>
<font color="Maroon" title="c2">x</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c4">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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