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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/>  for <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">z</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="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">r</font>) &amp; <font color="Olive" title="b4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">r</font>) holds <br/><span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b4">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">r</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"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b4">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</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">  for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">z</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="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c2">r</font>) &amp; <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c2">r</font>) holds <br/><span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Olive" title="b1">y</font>,<font color="Olive" title="b3">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c2">r</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"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b3">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">z</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>) &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>) implies <span class="p1">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</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"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</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">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</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"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_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="E1:48_1"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> or <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</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">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</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"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T29" title="ENUMSET1:th.29">ENUMSET1:29</a></span>;<br/><a NAME="E3:48_1_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 <a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="topreal9.html#T29" target="_self" title="TOPREAL9:th.29">Th29</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_1"/>
 for <font color="Olive" title="b1">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">m</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="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> holds <br/><font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <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="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c6">m</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="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E5:48_1_1"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <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="c6">m</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="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span><br/><a NAME="E6:48_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_1_1"/>
<font color="Maroon" title="c6">m</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="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E3:48_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E5:48_1_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_2"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">z</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">(<span class="default"><a href="topreal9.html#K4" title="TOPREAL9:func.4">halfline</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</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"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <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="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c6">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c6">m</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="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:48_1_2_1"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<font color="Maroon" title="c6">m</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="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><a NAME="E2:48_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">w</font> = <font color="Maroon" title="c6">m</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a></span>;<br/><a NAME="E4:48_1_2_1"/>
<font color="Maroon" title="c6">m</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">y</font>,<font color="Maroon" title="c5">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">R</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E6:48_1_2_1"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon" title="c6">m</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="c8">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</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="c8">R</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">z</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:48_1_2_1"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
 <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="c8">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T26" target="_self" title="TOPREAL9:th.26">Th26</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">R</font> = <font color="Maroon" title="c8">R</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_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="E1:48_1_2_1_1"/>
( <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( <font color="Maroon" title="c9">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c9">R</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or <font color="Maroon" title="c9">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_2_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_2_1_1_1"/>
<font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:48_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_2_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="topreal9.html#T4" target="_self" title="TOPREAL9:th.4">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_2_1_1_1"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_2_1_1_2"/>
<font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:48_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_2_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="topreal9.html#T4" target="_self" title="TOPREAL9:th.4">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_2_1_1_2"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_2_1_1_3"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
( <font color="Maroon" title="c9">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c9">R</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E36">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_2_1_1_3_1"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_2_1_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_2_1_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E6:48_1_2_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E1:48_1_2_1_1_3"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="topreal9.html#T4" target="_self" title="TOPREAL9:th.4">Th4</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><span class="kw">end;</span></div><a NAME="E3:48_1_2_1_1_3"/>
<font color="Maroon" title="c7">w</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="c3">y</font>,<font color="Maroon" title="c5">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_2_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E1:48_1_2_1_1_3"><span class="lab"><font color="Green" title="E35">A12</font></span></a></span>;<br/><a NAME="E4:48_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_2_1_1_3"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E5:48_1_2_1_1_3"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="topreal9.html#T34" target="_self" title="TOPREAL9:th.34">Th34</a></span>;<br/><a NAME="E6:48_1_2_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_2_1_1_3"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="topreal9.html#T7" target="_self" title="TOPREAL9:th.7">Th7</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_1_2_1_1_3"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_2_1"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="topreal9.html#T9" target="_self" title="TOPREAL9:th.9">Th9</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:48_1_2_1_1_4"/><span class="lab"><font color="Green" title="E35">A15</font></span>: 
<font color="Maroon" title="c9">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:48_1_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A16</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T212" title="XREAL_1:th.212">XREAL_1:212</a></span>;<br/><span class="lab"><font color="Green" title="E37">A17</font></span>: <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> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</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="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span> = 
<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> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</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"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_2_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="euclid.html#T32" title="EUCLID:th.32">EUCLID:32</a></span>
<br/>.= 
<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> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</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"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></span>
<br/>.= 
<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> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</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"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2_1_1_4"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>
<br/>.= 
<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> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</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"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</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> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T26" title="EUCLID:th.26">EUCLID:26</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> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</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> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2_1_1_4"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>
<br/>.= 
<font color="Maroon" title="c5">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T27" title="EUCLID:th.27">EUCLID:27</a></span>
;<br/><div><span class="lab"><font color="Green" title="E38">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1_1_4_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_2_1_1_4_2"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_2_1_1_4_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_2_1_1_4_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E2:48_1_2_1_1_4"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="txt" href="#E3:48_1_2_1_1_4"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="ref" href="topreal9.html#T4" target="_self" title="TOPREAL9:th.4">Th4</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><span class="kw">end;</span></div><a NAME="E5:48_1_2_1_1_4"/>
<font color="Maroon" title="c5">z</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="c3">y</font>,<font color="Maroon" title="c7">w</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2_1_1_4"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="txt" href="#E2:48_1_2_1_1_4"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="txt" href="#E3:48_1_2_1_1_4"><span class="lab"><font color="Green" title="E37">A17</font></span></a></span>;<br/><a NAME="E6:48_1_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A19</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">w</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_2_1_1_4"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E7:48_1_2_1_1_4"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">w</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:48_1_2_1"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="topreal9.html#T34" target="_self" title="TOPREAL9:th.34">Th34</a></span>;<br/><a NAME="E8:48_1_2_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_2_1_1_4"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="ref" href="topreal9.html#T7" target="_self" title="TOPREAL9:th.7">Th7</a></span>;<br/><span class="kw">hence </span><a NAME="E9:48_1_2_1_1_4"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="topreal9.html#T9" target="_self" title="TOPREAL9:th.9">Th9</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c6">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c6">m</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="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E3:48_1_2"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <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="c6">m</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="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)</span>)</span></span><br/><a NAME="E4:48_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A20</font></span>: 
( <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E5:48_1_2"/>
( <font color="Maroon" title="c3">y</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">y</font>,<font color="Maroon" title="c5">z</font>) &amp; <font color="Maroon" title="c5">z</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">y</font>,<font color="Maroon" title="c5">z</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T27" target="_self" title="TOPREAL9:th.27">Th27</a>, <a class="ref" href="topreal9.html#T28" target="_self" title="TOPREAL9:th.28">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_2"/>
<font color="Maroon" title="c6">m</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="c3">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">r</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E4:48_1_2"><span class="lab"><font color="Green" title="E31">A20</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
