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<span class="kw">let </span><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">q1</font>, <font color="Maroon" title="c4">q2</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> 3; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c3">q1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c4">q2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c3">q1</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c4">q2</font></span>)</span></span><br/>







<a NAME="E1:105"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#T87" target="_self" title="EUCLID_8:th.87">Th138</a></span>;<br/>
<a NAME="E2:105"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c3">q1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c4">q2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="euclid_8.html#T86" target="_self" title="EUCLID_8:th.86">Th137</a></span>;<br/>
<a NAME="E3:105"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c3">q1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c4">q2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c3">q1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c4">q2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="euclid_8.html#T86" target="_self" title="EUCLID_8:th.86">Th137</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:105"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c3">q1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c4">q2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c3">q1</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_8.html#K5" title="EUCLID_8:func.5">&lt;X&gt;</a> <font color="Maroon" title="c4">q2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</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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