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<span class="kw">let </span><font color="Maroon" title="c1">r1</font>, <font color="Maroon" title="c2">r2</font>, <font color="Maroon" title="c3">r19</font>, <font color="Maroon" title="c4">r29</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">r2</font> &amp; <font color="Maroon" title="c3">r19</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">r29</font> implies <span class="p1"><a href="sppol_2.html#K1" title="SPPOL_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font>,<font color="Maroon" title="c4">r29</font></span><a href="sppol_2.html#K1" title="SPPOL_2:func.1">.]</a></span> is  <a href="sppol_2.html#NV1" title="SPPOL_2:NV.1">special_polygonal</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:89"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
<font color="Maroon" title="c1">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">r2</font>
 <span class="kw">and </span><br/><a NAME="E2:89"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
<font color="Maroon" title="c3">r19</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">r29</font>
 ; <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="sppol_2.html#K1" title="SPPOL_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font>,<font color="Maroon" title="c4">r29</font></span><a href="sppol_2.html#K1" title="SPPOL_2:func.1">.]</a></span> is  <a href="sppol_2.html#NV1" title="SPPOL_2:NV.1">special_polygonal</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">p1</font> = <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">p2</font> = <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">p3</font> = <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">p4</font> = <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>;<br/>
<a NAME="E3:89"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<span class="kw">take </span>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="sppol_2.html#D2" target="_self" title="SPPOL_2:def.2">SPPOL_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p2</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> 2</span>)</span> st <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<font color="Olive" title="b1">p2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <span class="p1"><a href="sppol_2.html#K1" title="SPPOL_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font>,<font color="Maroon" title="c4">r29</font></span><a href="sppol_2.html#K1" title="SPPOL_2:func.1">.]</a></span></span><br/>

<span class="kw">take </span>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
; <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.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <span class="p1"><a href="sppol_2.html#K1" title="SPPOL_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font>,<font color="Maroon" title="c4">r29</font></span><a href="sppol_2.html#K1" title="SPPOL_2:func.1">.]</a></span></span><br/>

<span class="kw">thus </span><a NAME="E4:89"/>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="finseq_1.html#T77" title="FINSEQ_1:th.77">FINSEQ_1:77</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="sppol_2.html#D1" target="_self" title="SPPOL_2:def.1">SPPOL_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a> st <br/>( <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f1</font></span>)</span> &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="sppol_2.html#K1" title="SPPOL_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font>,<font color="Maroon" title="c4">r29</font></span><a href="sppol_2.html#K1" title="SPPOL_2:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b2">f2</font></span>)</span> )</span><br/>

<a NAME="E5:89"/><span class="lab"><font color="Green" title="E70">A4</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E6:89"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r29</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E7:89"/><span class="lab"><font color="Green" title="E72">A6</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">f1</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">f2</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>;<br/>
<a NAME="E8:89"/><span class="lab"><font color="Green" title="E73">A7</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E9:89"/><span class="lab"><font color="Green" title="E74">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<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="E10:89"/><span class="lab"><font color="Green" title="E75">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<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="E11:89"/><span class="lab"><font color="Green" title="E76">A10</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">f1</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>, <font color="Maroon" title="c12">f2</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> as   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="ref" href="topreal3.html#T34" title="TOPREAL3:th.34">TOPREAL3:34</a>, <a class="ref" href="topreal3.html#T35" title="TOPREAL3:th.35">TOPREAL3:35</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">f1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f2</font> being   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a> st <br/>( <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f1</font></span>)</span> &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="sppol_2.html#K1" title="SPPOL_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font>,<font color="Maroon" title="c4">r29</font></span><a href="sppol_2.html#K1" title="SPPOL_2:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">f2</font></span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c12">f2</font>
; <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.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f1</font></span>)</span> &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="sppol_2.html#K1" title="SPPOL_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font>,<font color="Maroon" title="c4">r29</font></span><a href="sppol_2.html#K1" title="SPPOL_2:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f2</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E13:89"/>
( <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f1</font></span>)</span> &amp; <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:89"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="txt" href="#E9:89"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="ref" href="finseq_4.html#T18" title="FINSEQ_4:th.18">FINSEQ_4:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="sppol_2.html#K1" title="SPPOL_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font>,<font color="Maroon" title="c4">r29</font></span><a href="sppol_2.html#K1" title="SPPOL_2:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f2</font></span>)</span> )</span><br/>

<a NAME="E14:89"/><span class="lab"><font color="Green" title="E77">A11</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></span>;<br/>
<a NAME="E15:89"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:89"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p4"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="txt" href="#E11:89"><span class="lab"><font color="Green" title="E76">A10</font></span></a>, <a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="txt" href="#E7:89"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="topreal3.html#T27" title="TOPREAL3:th.27">TOPREAL3:27</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="ref" href="topreal3.html#T28" title="TOPREAL3:th.28">TOPREAL3:28</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c3">r19</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c4">r29</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a></span>
;<br/> <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="sppol_2.html#K1" title="SPPOL_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font>,<font color="Maroon" title="c4">r29</font></span><a href="sppol_2.html#K1" title="SPPOL_2:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f2</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E17:89"/>
<span class="p1"><a href="sppol_2.html#K1" title="SPPOL_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font>,<font color="Maroon" title="c3">r19</font>,<font color="Maroon" title="c4">r29</font></span><a href="sppol_2.html#K1" title="SPPOL_2:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:89"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="ref" href="topreal3.html#T16" title="TOPREAL3:th.16">TOPREAL3:16</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/>


</div>
