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<span class="kw">let </span><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c3">z1</font>, <font color="Maroon" title="c4">x2</font>, <font color="Maroon" title="c5">y2</font>, <font color="Maroon" title="c6">z2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font>,<font color="Maroon" title="c3">z1</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">z2</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span><br/>











<a NAME="E1:38"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<a NAME="E2:38"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<a NAME="E3:38"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<a NAME="E4:38"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font>,<font color="Maroon" title="c3">z1</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<a NAME="E5:38"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font>,<font color="Maroon" title="c3">z1</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<a NAME="E6:38"/>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font>,<font color="Maroon" title="c3">z1</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:38"/>
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font>,<font color="Maroon" title="c3">z1</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">z2</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="euclid_8.html#T55" title="EUCLID_8:th.55">EUCLID_8:55</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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