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<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s2</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">t1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">t2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">d</font>,<font color="Maroon">e</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">h</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/>

<a NAME="E2:7"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E3:7"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:7"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E5:7"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<i><font color="Green">E45</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>
;<br/>
<a NAME="E7:7"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E8:7"/><i><font color="Green">E47</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E9:7"/><i><font color="Green">E48</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">d</font>,<font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">e</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E10:7"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">s1</font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <b>by </b><i>, , , <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Bb</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">s1</font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <b>by </b><i>, , , <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<b>assume </b><a NAME="E13:7"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">s1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">f3</font> = <font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Bb</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">d</font>,<font color="Maroon">e</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/>
<a NAME="E15:7"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">f3</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">e</font></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E17:7"/><i><font color="Green">E77</font></i>: 
<a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">d</font>,<font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T26">TOPMETR:26</a></i>;<br/>
<a NAME="E18:7"/><i><font color="Green">E78</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">s2</font>,<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E19:7"/><b>then </b>
<font color="Maroon">f3</font> is   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">s2</font>,<font color="Maroon">s1</font></span>)</span>,<a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T4">JORDAN6:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">s2</font>,<font color="Maroon">s1</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E21:7"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">s2</font>,<font color="Maroon">s1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E22:7"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">s1</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E23:7"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, , , <a class="ref" href="jordan6.html#T4">JORDAN6:4</a></i>;<br/>
<b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E24:7"/>
<font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E25:7"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E26:7"/>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E27:7"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E28:7"/>
<font color="Maroon">t1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t2</font>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E29:7"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">t1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t2</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E30:7"/><b>then </b>
<font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E31:7"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">t1</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E32:7"/>
<font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E33:7"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E35:7"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s1</font> )
 <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E1"><i><font color="Green">Lemma26</font></i></a>, <a class="ref" href="jgraph_5.html#T2" target="_self">Th2</a>, , <a class="ref" href="topreal5.html#T13">TOPREAL5:13</a></i>;<br/>
<a NAME="E36:7"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E2:7"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E37:7"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E2:7"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E38:7"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:7"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <b>by </b><i>, , <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B1b</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <b>by </b><i>, , <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f4</font> = <font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1b</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B1</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">d</font>,<font color="Maroon">e</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/>
<a NAME="E42:7"/><i><font color="Green">E94</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">s1</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E43:7"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">f4</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E44:7"/>
<font color="Maroon">f4</font> is   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">s1</font>,<font color="Maroon">b</font></span>)</span>,<a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E5:7"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan6.html#T4">JORDAN6:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">s1</font>,<font color="Maroon">b</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E46:7"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E5:7"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan6.html#T4">JORDAN6:4</a></i>;<br/>
<a NAME="E47:7"/>
<font color="Maroon">s2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E48:7"/><b>then </b>
<font color="Maroon">t2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1"><i><font color="Green">Lemma26</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E49:7"/><b>then </b><i><font color="Green">E98</font></i>: 
( <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> )
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E50:7"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">s1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E51:7"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E52:7"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E53:7"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E54:7"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E55:7"/>
( <font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">t1</font> )
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E57:7"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">s1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , , , <a class="ref" href="topreal5.html#T12">TOPREAL5:12</a></i>;<br/>
<a NAME="E58:7"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E59:7"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E60:7"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E61:7"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">s1</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E2:7"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then </b><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t2</font></span>)</span> <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="txt" href="jgraph_5.html#E2:7"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E6:7"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E63:7"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E2:7"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green">E33</font></i></a>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div>
