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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<i><font color="Green">E44</font></i>:  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="pscomp_1.html#K6">Cl</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E2:56"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal6.html#T80">TOPREAL6:80</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="pscomp_1.html#T34">PSCOMP_1:34</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E4:56"/><i><font color="Green">E45</font></i>: 
 <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>;<br/>
<a NAME="E5:56"/><i><font color="Green">E46</font></i>: 
 <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>;<br/>
<a NAME="E6:56"/><i><font color="Green">E47</font></i>: 
 <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>;<br/>
<a NAME="E7:56"/><i><font color="Green">E48</font></i>: 
 <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>;<br/>
<a NAME="E8:56"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E4:56"><i><font color="Green">E45</font></i></a>, <a class="txt" href="rcomp_3.html#E6:56"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a></i>;<br/>
<i><font color="Green">E50</font></i>:  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
 <a href="pscomp_1.html#K6">Cl</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E2:56"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal6.html#T80">TOPREAL6:80</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K6">Cl</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6">Cl</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E8:56"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rcomp_3.html#E4:56"><i><font color="Green">E45</font></i></a>, <a class="txt" href="rcomp_3.html#E6:56"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rcomp_3.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p3">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6">Cl</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E4:56"><i><font color="Green">E45</font></i></a>, <a class="ref" href="topreal6.html#T80">TOPREAL6:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p3">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E6:56"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topreal6.html#T80">TOPREAL6:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E4:56"><i><font color="Green">E45</font></i></a>, <a class="txt" href="rcomp_3.html#E7:56"><i><font color="Green">E48</font></i></a>, <a class="ref" href="borsuk_5.html#T77">BORSUK_5:77</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E5:56"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rcomp_3.html#E6:56"><i><font color="Green">E47</font></i></a>, <a class="ref" href="borsuk_5.html#T75">BORSUK_5:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E5:56"><i><font color="Green">E46</font></i></a>, <a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>
;<br/>
<a NAME="E10:56"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:56"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rcomp_3.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E11:56"/>
 <a href="tops_1.html#K2">Fr</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E3:56"><i><font color="Green">E44</font></i></a>, <a class="txt" href="rcomp_3.html#E9:56"><i><font color="Green">E50</font></i></a>, <a class="ref" href="tops_1.html#D2">TOPS_1:def 2</a></i>;<br/>


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