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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">e</font>, <font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</font>, <font color="Maroon" title="c7">t1</font>, <font color="Maroon" title="c8">t2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">h</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">d</font>,<font color="Maroon" title="c4">e</font></span>)</span>  st <font color="Olive" title="b1">h</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t1</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t2</font> &amp; <font color="Olive" title="b1">h</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">d</font> &amp; <font color="Olive" title="b1">h</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">e</font> &amp; <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c7">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">t2</font> &amp; <font color="Maroon" title="c5">s1</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> &amp; <font color="Maroon" title="c6">s2</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> holds <br/><font color="Maroon" title="c5">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">s2</font></span><br/><b>let </b><font color="Maroon" title="c9">h</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">d</font>,<font color="Maroon" title="c4">e</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">h</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t1</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t2</font> &amp; <font color="Maroon" title="c9">h</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">d</font> &amp; <font color="Maroon" title="c9">h</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">e</font> &amp; <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c7">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">t2</font> &amp; <font color="Maroon" title="c5">s1</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> &amp; <font color="Maroon" title="c6">s2</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> implies <font color="Maroon" title="c5">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">s2</font> )</span><br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c9">h</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t1</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t2</font> &amp; <font color="Maroon" title="c9">h</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">d</font> &amp; <font color="Maroon" title="c9">h</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">e</font> &amp; <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c7">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">t2</font> &amp; <font color="Maroon" title="c5">s1</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> &amp; <font color="Maroon" title="c6">s2</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">s2</font></span><br/>

<a NAME="E2:7"/><b>then </b><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c9">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E3:7"/><i><font color="Green" title="E9">A3</font></i>: 
( <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">s2</font> &amp; <font color="Maroon" title="c6">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:7"/><i><font color="Green" title="E10">A4</font></i>: 
( <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">s1</font> &amp; <font color="Maroon" title="c5">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<i><font color="Green" title="E11">A5</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h</font> = 
 <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>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<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>
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
;<br/>
<a NAME="E6:7"/><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c9">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:7"/><i><font color="Green" title="E13">A7</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">a</font>,<font color="Maroon" title="c2">b</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">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:7"/><i><font color="Green" title="E14">A8</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="c3">d</font>,<font color="Maroon" title="c4">e</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="c3">d</font>,<font color="Maroon" title="c4">e</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E9:7"/><i><font color="Green" title="E15">A9</font></i>: 
<font color="Maroon" title="c9">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">B</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c5">s1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <b>by </b><i><a class="txt" href="jgraph_5.html#E3:7"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E7:7"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="rcomp_1.html#T49" title="RCOMP_1:th.49">RCOMP_1:49</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">Bb</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c5">s1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <b>by </b><i><a class="txt" href="jgraph_5.html#E3:7"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E7:7"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="rcomp_1.html#T49" title="RCOMP_1:th.49">RCOMP_1:49</a></i>;<br/>
<b>assume </b><a NAME="E12:7"/><i><font color="Green" title="E16">A10</font></i>: 
<font color="Maroon" title="c5">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>reconsider </b><font color="Maroon" title="c12">f3</font> = <font color="Maroon" title="c9">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">Bb</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">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">d</font>,<font color="Maroon" title="c4">e</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></i>;<br/>
<a NAME="E14:7"/><i><font color="Green" title="E17">A11</font></i>: 
<font color="Maroon" title="c12">f3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E6:7"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">C</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">e</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E16:7"/><i><font color="Green" title="E18">A12</font></i>: 
<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">e</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="topmetr.html#T26" title="TOPMETR:th.26">TOPMETR:26</a></i>;<br/>
<a NAME="E17:7"/><i><font color="Green" title="E19">A13</font></i>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">B</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:7"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E12:7"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="topmetr.html#T30" title="TOPMETR:th.30">TOPMETR:30</a></i>;<br/>
<a NAME="E18:7"/><b>then </b>
<font color="Maroon" title="c12">f3</font> is   <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="c6">s2</font>,<font color="Maroon" title="c5">s1</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E16:7"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="jordan6.html#T4" title="JORDAN6:th.4">JORDAN6:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c9">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">B</font> as   <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="c6">s2</font>,<font color="Maroon" title="c5">s1</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/>
<a NAME="E20:7"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">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="c6">s2</font>,<font color="Maroon" title="c5">s1</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/>
<a NAME="E21:7"/><b>then </b><i><font color="Green" title="E20">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">f</font> <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="c6">s2</font>,<font color="Maroon" title="c5">s1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E12:7"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E22:7"/><i><font color="Green" title="E21">A15</font></i>: 
<font color="Maroon" title="c14">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E14:7"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="jgraph_5.html#E16:7"><i><font color="Green" title="E18">A12</font></i></a>, <a class="txt" href="jgraph_5.html#E17:7"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="jordan6.html#T4" title="JORDAN6:th.4">JORDAN6:4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">t</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">t1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/>
<a NAME="E23:7"/>
<font color="Maroon" title="c6">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E12:7"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E24:7"/><b>then </b><i><font color="Green" title="E22">A16</font></i>: 
<font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t2</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E25:7"/>
<font color="Maroon" title="c5">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E12:7"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E26:7"/><b>then </b><i><font color="Green" title="E23">A17</font></i>: 
<font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E27:7"/>
<font color="Maroon" title="c7">t1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">t2</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E2:7"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="jgraph_5.html#E5:7"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="jgraph_5.html#E12:7"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E28:7"/><b>then </b><i><font color="Green" title="E24">A18</font></i>: 
<font color="Maroon" title="c7">t1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">t2</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</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="E29:7"/><b>then </b>
<font color="Maroon" title="c7">t1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">t1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">t1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">t2</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E30:7"/><b>then </b><i><font color="Green" title="E25">A19</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">t1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/>
<a NAME="E31:7"/>
<font color="Maroon" title="c7">t1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">t2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">t2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">t2</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E28:7"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E32:7"/><b>then </b><i><font color="Green" title="E26">A20</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E34:7"/><i><font color="Green" title="E27">A21</font></i>: 
( <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp; <font color="Maroon" title="c6">s2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">r</font> &amp; <font color="Maroon" title="c16">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s1</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E12:7"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="jgraph_5.html#E22:7"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="jgraph_5.html#E24:7"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="jgraph_5.html#E26:7"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="jgraph_5.html#E30:7"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="topreal5.html#T13" title="TOPREAL5:th.13">TOPREAL5:13</a></i>;<br/>
<a NAME="E35:7"/><i><font color="Green" title="E28">A22</font></i>: 
<font color="Maroon" title="c16">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E34:7"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E36:7"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">r</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:7"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E34:7"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E37:7"/><b>then </b><i><font color="Green" title="E29">A23</font></i>: 
<font color="Maroon" title="c16">r</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>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E35:7"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">B1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <b>by </b><i><a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E7:7"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="rcomp_1.html#T49" title="RCOMP_1:th.49">RCOMP_1:49</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">B1b</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <b>by </b><i><a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E7:7"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="rcomp_1.html#T49" title="RCOMP_1:th.49">RCOMP_1:49</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">f4</font> = <font color="Maroon" title="c9">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c18">B1b</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">B1</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">d</font>,<font color="Maroon" title="c4">e</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></i>;<br/>
<a NAME="E41:7"/><i><font color="Green" title="E30">A24</font></i>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">B1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="topmetr.html#T30" title="TOPMETR:th.30">TOPMETR:30</a></i>;<br/>
<a NAME="E42:7"/><i><font color="Green" title="E31">A25</font></i>: 
<font color="Maroon" title="c19">f4</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E6:7"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/>
<a NAME="E43:7"/>
<font color="Maroon" title="c19">f4</font> is   <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="c5">s1</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E16:7"><i><font color="Green" title="E18">A12</font></i></a>, <a class="txt" href="jgraph_5.html#E41:7"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="jordan6.html#T4" title="JORDAN6:th.4">JORDAN6:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">f1</font> = <font color="Maroon" title="c9">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B1</font> as   <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="c5">s1</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/>
<a NAME="E45:7"/><i><font color="Green" title="E32">A26</font></i>: 
<font color="Maroon" title="c20">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E16:7"><i><font color="Green" title="E18">A12</font></i></a>, <a class="txt" href="jgraph_5.html#E41:7"><i><font color="Green" title="E30">A24</font></i></a>, <a class="txt" href="jgraph_5.html#E42:7"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="jordan6.html#T4" title="JORDAN6:th.4">JORDAN6:4</a></i>;<br/>
<a NAME="E46:7"/>
<font color="Maroon" title="c6">s2</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="c14">f</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E12:7"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="jgraph_5.html#E21:7"><i><font color="Green" title="E20">A14</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E47:7"/><b>then </b>
<font color="Maroon" title="c8">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f3</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:7"><i><font color="Green" title="E22">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E48:7"/><b>then </b><i><font color="Green" title="E33">A27</font></i>: 
( <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">t2</font> &amp; <font color="Maroon" title="c8">t2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">e</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E8:7"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E49:7"/><b>then </b><i><font color="Green" title="E34">A28</font></i>: 
<font color="Maroon" title="c5">s1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E28:7"><i><font color="Green" title="E24">A18</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="E50:7"/><i><font color="Green" title="E35">A29</font></i>: 
<font color="Maroon" title="c5">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E51:7"/><i><font color="Green" title="E36">A30</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E52:7"/><i><font color="Green" title="E37">A31</font></i>: 
<font color="Maroon" title="c20">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E50:7"><i><font color="Green" title="E35">A29</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E53:7"/><i><font color="Green" title="E38">A32</font></i>: 
<font color="Maroon" title="c20">f1</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">e</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E51:7"><i><font color="Green" title="E36">A30</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E54:7"/>
( <font color="Maroon" title="c4">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">t1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">t1</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E30:7"><i><font color="Green" title="E25">A19</font></i></a>, <a class="txt" href="jgraph_5.html#E32:7"><i><font color="Green" title="E26">A20</font></i></a>, <a class="txt" href="jgraph_5.html#E48:7"><i><font color="Green" title="E33">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E56:7"/><i><font color="Green" title="E39">A33</font></i>: 
( <font color="Maroon" title="c20">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp; <font color="Maroon" title="c5">s1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">r1</font> &amp; <font color="Maroon" title="c21">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E45:7"><i><font color="Green" title="E32">A26</font></i></a>, <a class="txt" href="jgraph_5.html#E49:7"><i><font color="Green" title="E34">A28</font></i></a>, <a class="txt" href="jgraph_5.html#E52:7"><i><font color="Green" title="E37">A31</font></i></a>, <a class="txt" href="jgraph_5.html#E53:7"><i><font color="Green" title="E38">A32</font></i></a>, <a class="ref" href="topreal5.html#T12" title="TOPREAL5:th.12">TOPREAL5:12</a></i>;<br/>
<a NAME="E57:7"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">r1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:7"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E56:7"><i><font color="Green" title="E39">A33</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E58:7"/><b>then </b><i><font color="Green" title="E40">A34</font></i>: 
<font color="Maroon" title="c21">r1</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>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E56:7"><i><font color="Green" title="E39">A33</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E59:7"/><i><font color="Green" title="E41">A35</font></i>: 
<font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E56:7"><i><font color="Green" title="E39">A33</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E60:7"/>
<font color="Maroon" title="c16">r</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="c6">s2</font>,<font color="Maroon" title="c5">s1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E34:7"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>then </b><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">r</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="txt" href="jgraph_5.html#E34:7"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">r1</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E56:7"><i><font color="Green" title="E39">A33</font></i></a>, <a class="txt" href="jgraph_5.html#E59:7"><i><font color="Green" title="E41">A35</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E62:7"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E5:7"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="jgraph_5.html#E9:7"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E34:7"><i><font color="Green" title="E27">A21</font></i></a>, <a class="txt" href="jgraph_5.html#E37:7"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="jgraph_5.html#E56:7"><i><font color="Green" title="E39">A33</font></i></a>, <a class="txt" href="jgraph_5.html#E58:7"><i><font color="Green" title="E40">A34</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
