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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="funct_1.html#V2">one-to-one</a> <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#R1">is_increasing_on</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#R2">is_decreasing_on</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green">E23</font></i>: 
 <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#R1">is_increasing_on</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#R2">is_decreasing_on</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="fcont_3.html#E1:29"><i><font color="Green">E22</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#R1">is_increasing_on</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="rfunct_2.html#R1">is_increasing_on</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="fcont_3.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E3:29_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="rfunct_2.html#R3">is_non_decreasing_on</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T55">RFUNCT_2:55</a></i>;<br/><a NAME="E4:29_1_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="rfunct_2.html#R5">is_monotone_on</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#D6">RFUNCT_2:def 6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:29_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:29_1_1_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T78">PARTFUN2:78</a></i>;<br/><a NAME="E4:29_1_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="fcont_3.html#E3:29_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:29_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E6:29_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E7:29_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:29_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/></div><b>end;</b></div><a NAME="E6:29_1_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
 <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="fcont_3.html#E2:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>hence </b><a NAME="E8:29_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fcont_3.html#E4:29_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fcont_3.html#E6:29_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fcont_3.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_2.html#R2">is_decreasing_on</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="rfunct_2.html#R2">is_decreasing_on</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="fcont_3.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:29_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="rfunct_2.html#R4">is_non_increasing_on</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T56">RFUNCT_2:56</a></i>;<br/><a NAME="E4:29_1_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="rfunct_2.html#R5">is_monotone_on</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#D6">RFUNCT_2:def 6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:29_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:29_1_1_2_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T78">PARTFUN2:78</a></i>;<br/><a NAME="E4:29_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fcont_3.html#E3:29_1_1_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:29_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E6:29_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="fcont_3.html#E3:29_1_1_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E7:29_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:29_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/></div><b>end;</b></div><a NAME="E6:29_1_1_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
 <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="fcont_3.html#E2:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>hence </b><a NAME="E8:29_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fcont_3.html#E4:29_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fcont_3.html#E6:29_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fcont_3.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:29"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div>
