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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">p3</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid_5.html#K6" title="EUCLID_5:func.6">|{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="euclid_5.html#K6" title="EUCLID_5:func.6">}|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K6" title="EUCLID_5:func.6">|{</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font></span><a href="euclid_5.html#K6" title="EUCLID_5:func.6">}|</a></span></span><br/>





<span class="p1"><a href="euclid_5.html#K6" title="EUCLID_5:func.6">|{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="euclid_5.html#K6" title="EUCLID_5:func.6">}|</a></span> = 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>,<span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></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="ref" href="euclid_5.html#T3" target="_self" title="EUCLID_5:th.3">Th3</a></span>
<br/>.= 
<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_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T30" target="_self" title="EUCLID_5:th.30">Th30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1: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_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>,<span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></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="ref" href="euclid_5.html#T30" target="_self" title="EUCLID_5:th.30">Th30</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">p2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid_5.html#K5" title="EUCLID_5:func.5">&lt;X&gt;</a> <font color="Maroon" title="c1">p1</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="ref" href="euclid_5.html#T3" target="_self" title="EUCLID_5:th.3">Th3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:35"/>
<span class="p1"><a href="euclid_5.html#K6" title="EUCLID_5:func.6">|{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="euclid_5.html#K6" title="EUCLID_5:func.6">}|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K6" title="EUCLID_5:func.6">|{</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font></span><a href="euclid_5.html#K6" title="EUCLID_5:func.6">}|</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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