<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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">p</font>, <font color="Olive" title="b2">q</font> being  <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> <font color="Maroon" title="c1">n</font></span>)</span> holds  <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b1">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Olive" title="b2">q</font>,<font color="Olive" title="b2">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</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> <font color="Maroon" title="c1">n</font></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="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span><br/>





<a NAME="E1:64"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</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="ref" href="euclid_2.html#T57" target="_self" title="EUCLID_2:th.57">Th57</a></span>;<br/>
<a NAME="E2:64"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</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="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E3:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</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="c2">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</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="ref" href="euclid_2.html#T53" target="_self" title="EUCLID_2:th.53">Th53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span>

;<br/>
<a NAME="E5:64"/><span class="kw">then </span>
2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_2.html#T57" target="_self" title="EUCLID_2:th.57">Th57</a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/>
<a NAME="E6:64"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/>
<a NAME="E7:64"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:64"/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c3">q</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</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/>


</div>
