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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">x0</font>, <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">y0</font>, <font color="Olive" title="b4">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><br/> for <font color="Olive" title="b5">L1</font>, <font color="Olive" title="b6">L2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">L1</font> &amp; <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">L1</font> &amp; <font color="Olive" title="b1">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">x1</font> &amp; <font color="Olive" title="b3">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">L2</font> &amp; <font color="Olive" title="b4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">L2</font> &amp; <font color="Olive" title="b3">y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">y1</font> &amp; <font color="Olive" title="b5">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b6">L2</font> holds <br/><font color="Olive" title="b2">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Olive" title="b1">x0</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Olive" title="b4">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Olive" title="b3">y0</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x0</font>, <font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">y0</font>, <font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L1</font>, <font color="Olive" title="b2">L2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font>  st <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">L1</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">L1</font> &amp; <font color="Maroon" title="c2">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Maroon" title="c4">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">L2</font> &amp; <font color="Maroon" title="c5">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">L2</font> &amp; <font color="Maroon" title="c4">y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Olive" title="b1">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b2">L2</font> holds <br/><font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x0</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">y0</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">L1</font>, <font color="Maroon" title="c7">L2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_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">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L1</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L1</font> &amp; <font color="Maroon" title="c2">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Maroon" title="c4">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">L2</font> &amp; <font color="Maroon" title="c5">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">L2</font> &amp; <font color="Maroon" title="c4">y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c6">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c7">L2</font> implies <font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x0</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">y0</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:87"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
( <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L1</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L1</font> )
 <span class="kw">and </span><br/><a NAME="E2:87"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x1</font>
 <span class="kw">and </span><br/><a NAME="E3:87"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
( <font color="Maroon" title="c4">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">L2</font> &amp; <font color="Maroon" title="c5">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">L2</font> )
 <span class="kw">and </span><br/><a NAME="E4:87"/><span class="lab"><font color="Green" title="E70">A4</font></span>: 
<font color="Maroon" title="c4">y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y1</font>
 <span class="kw">and </span><br/><a NAME="E5:87"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
<font color="Maroon" title="c6">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c7">L2</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="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x0</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">y0</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">x2</font>, <font color="Maroon" title="c9">x3</font>, <font color="Maroon" title="c10">y2</font>, <font color="Maroon" title="c11">y3</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><span class="kw"> such that </span><br/><a NAME="E7:87"/><span class="lab"><font color="Green" title="E72">A6</font></span>: 
<font color="Maroon" title="c6">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>
 <span class="kw">and </span><br/><a NAME="E8:87"/><span class="lab"><font color="Green" title="E73">A7</font></span>: 
<font color="Maroon" title="c7">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c10">y2</font>,<font color="Maroon" title="c11">y3</font>
 <span class="kw">and </span><br/><a NAME="E9:87"/><span class="lab"><font color="Green" title="E74">A8</font></span>: 
<font color="Maroon" title="c9">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x2</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="ref" href="euclidlp.html#D8" target="_self" title="EUCLIDLP:def.8">Def8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:87"/><span class="lab"><font color="Green" title="E75">A9</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E7:87"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="euclidlp.html#T36" target="_self" title="EUCLIDLP:th.36">Th36</a></span>;<br/>
<a NAME="E12:87"/>
<font color="Maroon" title="c9">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</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="#E9:87"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></span>;<br/>
<a NAME="E13:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A10</font></span>: 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D18" title="RVSUM_1:def.18">RVSUM_1:def 18</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E15:87"/><span class="lab"><font color="Green" title="E77">A11</font></span>: 
<font color="Maroon" title="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="ref" href="euclidlp.html#T36" target="_self" title="EUCLIDLP:th.36">Th36</a></span>;<br/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">y0</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> = 
<font color="Maroon" title="c12">s</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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">y0</font></span>)</span></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="#E11:87"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="ref" href="euclid_4.html#T26" title="EUCLID_4:th.26">EUCLID_4:26</a></span>
<br/>.= 
<font color="Maroon" title="c12">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">y2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:87"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="ref" href="euclid_4.html#T27" title="EUCLID_4:th.27">EUCLID_4:27</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E76">A10</font></span></a></span>
;<br/>
<a NAME="E17:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A12</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x0</font>,<font color="Maroon" title="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">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="ref" href="rvsum_1.html#D18" title="RVSUM_1:def.18">RVSUM_1:def 18</a></span>;<br/>
<a NAME="E18:87"/>
( <font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x0</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="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">y0</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="#E2:87"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:87"/>
<font color="Maroon" title="c3">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x0</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c5">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">y0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:87"><span class="lab"><font color="Green" title="E78">A12</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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