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<b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</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> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> implies  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> )</span><br/>





<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> or  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> )</span><br/>

<b>assume </b><a NAME="E2:12"/><i><font color="Green" title="E9">A2</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a></span><br/>

<a NAME="E3:12"/><b>then </b><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="ref" href="sppol_1.html#T36" title="SPPOL_1:th.36">SPPOL_1:36</a></i>;<br/>
<a NAME="E4:12"/>
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:12"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="jordan.html#E2:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="sppol_1.html#D2" title="SPPOL_1:def.2">SPPOL_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:12"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a>
 <b>by </b><i><a class="txt" href="jordan.html#E3:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="sppol_1.html#T36" title="SPPOL_1:th.36">SPPOL_1:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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