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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font>, <font color="Maroon" title="c5">e</font>, <font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c7">g</font>, <font color="Maroon" title="c8">h</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>  st <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Olive" title="b1">F</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font> holds <br/><font color="Olive" title="b1">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">F</font> be   <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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c9">F</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font> implies <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">and </span><br/><a NAME="E3:28"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">and </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">and </span><br/><a NAME="E5:28"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E6:28"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c9">F</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E7:28"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
( <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> )
 <span class="kw">and </span><br/><a NAME="E8:28"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">and </span><br/><a NAME="E9:28"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<a NAME="E10:28"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:28"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l1</font> where <font color="Olive" title="b1">l1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l1</font> &amp; <font color="Olive" title="b1">l1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E24">A5</font></span></a></span>;<br/>
<a NAME="E12:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<font color="Maroon" title="c6">f</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">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E13:28"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<a NAME="E14:28"/><span class="kw">then </span>
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E15:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c8">h</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="c3">c</font>,<font color="Maroon" title="c4">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<a NAME="E16:28"/>
<font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E17:28"/><span class="kw">then </span>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l1</font> where <font color="Olive" title="b1">l1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l1</font> &amp; <font color="Olive" title="b1">l1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>;<br/>
<a NAME="E18:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c5">e</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">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E19:28"/><span class="kw">then </span>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<a NAME="E20:28"/><span class="kw">then </span>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E21:28"/><span class="kw">then </span>
<font color="Maroon" title="c7">g</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="c3">c</font>,<font color="Maroon" title="c4">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<a NAME="E22:28"/><span class="kw">then </span>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:28"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>;<br/>
<a NAME="E23:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<a NAME="E24:28"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
( <font color="Maroon" title="c9">F</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c9">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E25:28"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">aa</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">aa</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="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> or <font color="Maroon" title="c10">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c9">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:28_1"/>
<font color="Maroon" title="c10">aa</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="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<a NAME="E3:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l1</font> where <font color="Olive" title="b1">l1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l1</font> &amp; <font color="Olive" title="b1">l1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">l</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:28_1"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<font color="Maroon" title="c10">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">and </span><br/><a NAME="E6:28_1"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">and </span><br/><a NAME="E7:28_1"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font>
 ;<br/>
<a NAME="E8:28_1"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E9:28_1"/>
<font color="Maroon" title="c11">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l1</font> where <font color="Olive" title="b1">l1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l1</font> &amp; <font color="Olive" title="b1">l1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28_1"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="txt" href="#E7:28_1"><span class="lab"><font color="Green" title="E38">A19</font></span></a></span>;<br/>
<a NAME="E10:28_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A21</font></span>: 
<font color="Maroon" title="c11">l</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="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E11:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E8:28_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a></span>;<br/>
<a NAME="E12:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E8:28_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="tops_2.html#T62" title="TOPS_2:th.62">TOPS_2:62</a></span>;<br/>
<a NAME="E13:28_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E14:28_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">l</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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
;<br/>
<a NAME="E15:28_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">l</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">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">l</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E17:28_1"/><span class="lab"><font color="Green" title="E41">A22</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E8:28_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></span>;<br/>
<a NAME="E18:28_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A23</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E8:28_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="txt" href="#E10:28_1"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>;<br/>
<a NAME="E19:28_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">e9</font> = <font color="Maroon" title="c5">e</font>, <font color="Maroon" title="c14">f9</font> = <font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c15">x9</font> = <font color="Maroon" title="c12">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E8:28_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="txt" href="#E10:28_1"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="txt" href="#E17:28_1"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">g9</font> = <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">e9</font>, <font color="Maroon" title="c17">h9</font> = <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f9</font>, <font color="Maroon" title="c18">l9</font> = <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x9</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> ;<br/>
<a NAME="E22:28_1"/>
<font color="Maroon" title="c18">l9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
;<br/>
<a NAME="E23:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">l9</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="c3">c</font>,<font color="Maroon" title="c4">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">gg</font> = <font color="Maroon" title="c16">g9</font>, <font color="Maroon" title="c20">hh</font> = <font color="Maroon" title="c17">h9</font>, <font color="Maroon" title="c21">ll</font> = <font color="Maroon" title="c18">l9</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<a NAME="E25:28_1"/><span class="lab"><font color="Green" title="E43">A24</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:28_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">ll</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">hh</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="jordan5a.html#T16" target="_self" title="JORDAN5A:th.16">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:28_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E7:28_1"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E8:28_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="txt" href="#E10:28_1"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="txt" href="#E17:28_1"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E26:28_1"/>
<font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:28_1_2"/>
<font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:28_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">gg</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c21">ll</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="jordan5a.html#T16" target="_self" title="JORDAN5A:th.16">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:28_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E6:28_1"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E8:28_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="txt" href="#E10:28_1"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="txt" href="#E17:28_1"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E27:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l1</font> where <font color="Olive" title="b1">l1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l1</font> &amp; <font color="Olive" title="b1">l1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28_1"><span class="lab"><font color="Green" title="E43">A24</font></span></a></span>;<br/>
<a NAME="E28:28_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A25</font></span>: 
<font color="Maroon" title="c12">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="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E29:28_1"/>
<font color="Maroon" title="c10">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E5:28_1"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E8:28_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="txt" href="#E10:28_1"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="txt" href="#E17:28_1"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:28_1"/>
<font color="Maroon" title="c10">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:28_1"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="txt" href="#E28:28_1"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E26:28"/>
<font color="Maroon" title="c9">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">aa</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> or <font color="Maroon" title="c10">aa</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="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_2"/>
<font color="Maroon" title="c10">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">aa</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="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:28_2"/><span class="lab"><font color="Green" title="E35">A26</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">F</font>
 <span class="kw">and </span><br/><a NAME="E4:28_2"/><span class="lab"><font color="Green" title="E36">A27</font></span>: 
<font color="Maroon" title="c11">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="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:28_2"/><span class="lab"><font color="Green" title="E37">A28</font></span>: 
<font color="Maroon" title="c10">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E6:28_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_2"><span class="lab"><font color="Green" title="E36">A27</font></span></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x9</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E8:28_2"/><span class="lab"><font color="Green" title="E38">A29</font></span>: 
<font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">and </span><br/><a NAME="E9:28_2"/><span class="lab"><font color="Green" title="E39">A30</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x9</font>
 <span class="kw">and </span><br/><a NAME="E10:28_2"/><span class="lab"><font color="Green" title="E40">A31</font></span>: 
<font color="Maroon" title="c12">x9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font>
 ;<br/>
<a NAME="E11:28_2"/>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_2"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="txt" href="#E8:28_2"><span class="lab"><font color="Green" title="E38">A29</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E12:28_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
;<br/>
<a NAME="E13:28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A32</font></span>: 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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="c3">c</font>,<font color="Maroon" title="c4">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">Fx</font> = <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">e1</font> = <font color="Maroon" title="c5">e</font>, <font color="Maroon" title="c15">f1</font> = <font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c12">x9</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E3:28_2"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="txt" href="#E8:28_2"><span class="lab"><font color="Green" title="E38">A29</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">gg</font> = <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">e1</font>, <font color="Maroon" title="c18">hh</font> = <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">f1</font>, <font color="Maroon" title="c19">FFx</font> = <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E13:28_2"><span class="lab"><font color="Green" title="E41">A32</font></span></a></span>;<br/>
<a NAME="E17:28_2"/><span class="lab"><font color="Green" title="E42">A33</font></span>: 
<font color="Maroon" title="c13">Fx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_2_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> or <font color="Maroon" title="c6">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_2_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">Fx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:28_2_1_1_1"/>
<font color="Maroon" title="c13">Fx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E8:28_2"><span class="lab"><font color="Green" title="E38">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_2_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">Fx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font></span><br/><div class="add"><a NAME="E2:28_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28_2"><span class="lab"><font color="Green" title="E38">A29</font></span></a>, <a class="txt" href="#E10:28_2"><span class="lab"><font color="Green" title="E40">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:28_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">hh</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c19">FFx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="jordan5a.html#T16" target="_self" title="JORDAN5A:th.16">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E4:28_2_1_1_2"/>
<font color="Maroon" title="c13">Fx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E18:28_2"/>
<font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">Fx</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_2_2_1"/>
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> or <font color="Maroon" title="c5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_2_2_1_1"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">Fx</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:28_2_2_1_1"/>
<font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">Fx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E8:28_2"><span class="lab"><font color="Green" title="E38">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_2_2_1_2"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">Fx</font></span><br/><div class="add"><a NAME="E2:28_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28_2"><span class="lab"><font color="Green" title="E38">A29</font></span></a>, <a class="txt" href="#E9:28_2"><span class="lab"><font color="Green" title="E39">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:28_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">gg</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">FFx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="jordan5a.html#T16" target="_self" title="JORDAN5A:th.16">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E4:28_2_2_1_2"/>
<font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">Fx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E27">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E19:28_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l1</font> where <font color="Olive" title="b1">l1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l1</font> &amp; <font color="Olive" title="b1">l1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">h</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:28_2"><span class="lab"><font color="Green" title="E42">A33</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:28_2"/>
<font color="Maroon" title="c10">aa</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="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_2"><span class="lab"><font color="Green" title="E37">A28</font></span></a>, <a class="txt" href="#E8:28_2"><span class="lab"><font color="Green" title="E38">A29</font></span></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E27:28"/>
<font color="Maroon" title="c9">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">f</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c7">g</font>,<font color="Maroon" title="c8">h</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
