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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:6"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> is <a href="connsp_1.html#V2">connected</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">L</font> =  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1"/>
not  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 ;<br/><b>then consider </b><font color="Maroon">x0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">p0</font> = <font color="Maroon">x0</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E5:6_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E6:6_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/><a NAME="E7:6_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p0</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p0</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><i><font color="Green">E47</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">L</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><b>then reconsider </b><font color="Maroon">R1</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p0</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">L</font></span>)</span> <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, </i>;<br/><a NAME="E10:6_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E11:6_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p0</font>,<font color="Maroon">p2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/><a NAME="E12:6_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p0</font>,<font color="Maroon">p2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p0</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><b>then reconsider </b><font color="Maroon">R2</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p0</font>,<font color="Maroon">p2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">L</font></span>)</span> <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, </i>;<br/><b>reconsider </b><font color="Maroon">PX</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">L</font></span>)</span> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">PX1</font> =  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p0</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">L</font></span>)</span> <b>by </b><i>, , , <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/><b>reconsider </b><font color="Maroon">PX2</font> =  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p0</font>,<font color="Maroon">p2</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">L</font></span>)</span> <b>by </b><i>, , , <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/><a NAME="E17:6_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">L</font> is <a href="compts_1.html#V3">being_T2</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T3">TOPMETR:3</a></i>;<br/><a NAME="E18:6_1"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p0</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="sppol_1.html#T28">SPPOL_1:28</a></i>;<br/><a NAME="E19:6_1"/><b>then </b>
<font color="Maroon">PX1</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T42">BORSUK_1:42</a></i>;<br/><a NAME="E20:6_1"/><b>then </b>
<font color="Maroon">PX1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T16">COMPTS_1:16</a>, </i>;<br/><a NAME="E21:6_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">PX1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PX1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><a NAME="E22:6_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">PX</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T15">JORDAN2C:15</a>, </i>;<br/><a NAME="E23:6_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">R2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PX2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E24:6_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">R1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PX1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E25:6_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">PX1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E26:6_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">R2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PX1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">R2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E27:6_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">PX1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">PX2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#T14">TOPREAL1:14</a></i>;<br/><font color="Maroon">PX1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">R2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">PX1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">PX2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
;<br/><a NAME="E29:6_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E30:6_1"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">R2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E31:6_1"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p0</font>,<font color="Maroon">p2</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="sppol_1.html#T28">SPPOL_1:28</a></i>;<br/><a NAME="E32:6_1"/><b>then </b>
<font color="Maroon">PX2</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T42">BORSUK_1:42</a></i>;<br/><a NAME="E33:6_1"/><b>then </b>
<font color="Maroon">PX2</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T16">COMPTS_1:16</a>, </i>;<br/><a NAME="E34:6_1"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">PX2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PX2</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><a NAME="E35:6_1"/>
<font color="Maroon">R2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PX2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E36:6_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R2</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">PX2</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E37:6_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">R1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">R1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">PX2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E38:6_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">PX1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">PX2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#T14">TOPREAL1:14</a></i>;<br/><font color="Maroon">R1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">PX2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">PX1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">PX2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
;<br/><a NAME="E40:6_1"/><b>then </b>
<font color="Maroon">R1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E41:6_1"/><b>then </b>
<font color="Maroon">R1</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E42:6_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">R1</font>,<font color="Maroon">R2</font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i>, <a class="ref" href="connsp_1.html#D1">CONNSP_1:def 1</a></i>;<br/><a NAME="E43:6_1"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_4"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:6_1_4"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_4"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p0</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E44:6_1"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p0</font>,<font color="Maroon">p2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_5"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:6_1_5"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_5"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p0</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E84</font></i>: <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p0</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p0</font>,<font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="topreal1.html#T11">TOPREAL1:11</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p0</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p0</font>,<font color="Maroon">p2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p0</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p0</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">PX2</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<font color="Maroon">R1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">PX2</font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">R2</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
;<br/><a NAME="E46:6_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_7"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:6_1_7"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:6_1_7"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:6_1_7"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_8"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R1</font>
 ;<br/><a NAME="E2:6_1_8"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font>
 <b>by </b><i/>;<br/><a NAME="E3:6_1_8"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PX2</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E4:6_1_8"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PX1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">PX2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:6_1_8"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_9"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R2</font>
 ;<br/><a NAME="E2:6_1_9"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font>
 <b>by </b><i/>;<br/><a NAME="E3:6_1_9"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PX1</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E4:6_1_9"/><b>then </b>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PX1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">PX2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:6_1_9"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E49:6_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="connsp_1.html#T17">CONNSP_1:17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>, </i>;<br/>


</div>
