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<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">r</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:()"><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="c1">p</font>,<font color="Maroon" title="c2">q</font> is  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>  &amp; <font color="Maroon" title="c3">r</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="c1">p</font>,<font color="Maroon" title="c2">q</font> implies <font color="Maroon" title="c1">p</font> <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="c3">r</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:71"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> is  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">r</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="c1">p</font>,<font color="Maroon" title="c2">q</font> or <font color="Maroon" title="c1">p</font> <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="c3">r</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>

<a NAME="E2:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c1">p</font> <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">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_1.html#T37" target="_self" title="SPPOL_1:th.37">Th37</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:71"/>
<font color="Maroon" title="c3">r</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="c1">p</font>,<font color="Maroon" title="c2">q</font>
 ; <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="c1">p</font> <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="c3">r</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:71"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">q</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:71"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t</font>
 <span class="kw">and </span><br/><a NAME="E7:71"/>
<font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </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="c4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></span>
<br/>.= 
<font color="Maroon" title="c3">r</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="topreal3.html#T7" title="TOPREAL3:th.7">TOPREAL3:7</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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