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<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">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</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>  st <font color="Olive" title="b3">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Olive" title="b4">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> holds <br/> <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</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"> ( <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> implies  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><br/>

<a NAME="E3:16"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</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> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font></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> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c3">p2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="topreal1.html#T11" target="_self" title="TOPREAL1:th.11">Th11</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1_1"/>
( <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> or <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c3">p2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_1_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><br/><div class="add"><a NAME="E2:16_1_1_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="topreal1.html#E12"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a></span>;<br/><a NAME="E3:16_1_1_1"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="topreal1.html#E12"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:16_1_1_1"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_1_2"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><br/><div class="add"><a NAME="E2:16_1_1_2"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="topreal1.html#E12"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a></span>;<br/><a NAME="E3:16_1_1_2"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="topreal1.html#E12"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:16_1_1_2"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_2"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:16"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ; <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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