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<b>let </b><font color="Maroon" title="c1">ra</font>, <font color="Maroon" title="c2">rb</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>







<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c1">ra</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">rb</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">f</font> be  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/><b>let </b><font color="Maroon" title="c6">d</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>assume </b><a NAME="E2:7"/><i><font color="Green" title="E12">A2</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ra</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">rb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c6">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">b</font> )
 ;<br/>

<a NAME="E3:7"/><i><font color="Green" title="E13">A3</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="topreal5.html#E1:7"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="topreal5.html#T6" target="_self" title="TOPREAL5:th.6">Th6</a></i>;<br/>
<a NAME="E4:7"/><i><font color="Green" title="E14">A4</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E5:7"/><i><font color="Green" title="E15">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="topreal5.html#E1:7"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E6:7"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span>)</span></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="topreal5.html#E3:7"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="topreal5.html#E7"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/>
<a NAME="E7:7"/><i><font color="Green" title="E17">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1"/><i><font color="Green" title="E18">A8</font></i>: 
for <font color="Olive" title="b1">rc</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">ra</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">rc</font> &amp; <font color="Olive" title="b1">rc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">rb</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">rc</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d</font>
 ;<br/><div><i><font color="Green" title="E19">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_1"/>
<font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:7_1_1"/><i><font color="Green" title="E20">A10</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span>)</span> &amp; <font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:7_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="topreal5.html#E5:7"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="topreal5.html#E3:7_1_1"><i><font color="Green" title="E20">A10</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c7">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E6:7_1_1"/><i><font color="Green" title="E21">A11</font></i>: 
( <font color="Maroon" title="c1">ra</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">rb</font> )
 <b>by </b><i><a class="txt" href="topreal5.html#E5:7"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="topreal5.html#E3:7_1_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="topreal5.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E7:7_1_1"/><b>then </b><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c1">ra</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="topreal5.html#E2:7"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="topreal5.html#E3:7_1_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E8:7_1_1"/>
<font color="Maroon" title="c2">rb</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="topreal5.html#E2:7"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="topreal5.html#E3:7_1_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="topreal5.html#E6:7_1_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:7_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal5.html#E1:7_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="topreal5.html#E3:7_1_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="topreal5.html#E7:7_1_1"><i><font color="Green" title="E22">A12</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:7_1"/>
<font color="Maroon" title="c1">ra</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="topreal5.html#E1:7"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/><a NAME="E4:7_1"/><b>then </b><i><font color="Green" title="E20">A13</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topreal5.html#E2:7"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="topreal5.html#E4:7"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="topreal5.html#E5:7"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="topreal5.html#E7:7"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:7_1"/>
<font color="Maroon" title="c2">rb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="topreal5.html#E1:7"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/><a NAME="E6:7_1"/><b>then </b>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">ra</font>,<font color="Maroon" title="c2">rb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topreal5.html#E2:7"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="topreal5.html#E4:7"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="topreal5.html#E5:7"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="topreal5.html#E7:7"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E7:7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal5.html#E2:7"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="topreal5.html#E6:7"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="topreal5.html#E2:7_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="topreal5.html#E4:7_1"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="topreal5.html#T9" target="_self" title="TOPREAL5:th.9">Th9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:7"/>
 ex <font color="Olive" title="b1">rc</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">rc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c1">ra</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">rc</font> &amp; <font color="Olive" title="b1">rc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">rb</font> )
 ;<br/>


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