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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font> holds   <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Olive" title="b2">p2</font>,<font color="Olive" title="b1">p1</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</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="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span></span><br/>



<span class="kw">thus </span><a NAME="E1:80"/>
 <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="80_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) or <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:80_1"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:80_1"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c4">e</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p1</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="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p2</font></span>)</span>
 ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="80_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) ) or ( <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>) ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:80_1_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:80_1_1_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:80_1_1_1_1"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</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="#E3:80_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="80_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:80_1_1_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span><br/><div class="add"><a NAME="E2:80_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A2</font></span>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E3:80_1_1_1_2"/>
<font color="Maroon" title="c4">e</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:80_1_1_1_2"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1_1_2"><span class="lab"><font color="Green" title="E39">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E5:80_1"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E2:80"/>
(  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) &amp;  <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</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="ref" href="topreal9.html#T64" target="_self" onmouseover="rs('topreal9/T64')" onmouseout="rh()">Th62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:80"/>
<span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K5" title="RLTOPSP1:func.5">Line</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="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
