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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">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_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="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c7">t1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">h</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_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="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c7">t1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c9">h</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E2:11"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:11"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:11"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<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="c3">d</font>
 <span class="kw">and </span><br/><a NAME="E5:11"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">and </span><br/><a NAME="E6:11"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c7">t1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">t2</font>
 <span class="kw">and </span><br/><a NAME="E7:11"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:11"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E9:11"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E11:11"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
<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="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="topmetr.html#T26" title="TOPMETR:th.26">TOPMETR:26</a></span>;<br/>
<a NAME="E12:11"/><span class="lab"><font color="Green" title="E18">A11</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">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E13:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A12</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E16">A9</font></span></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></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:11"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="xxreal_1.html#T34" title="XXREAL_1:th.34">XXREAL_1:34</a></span>;<br/>
<span class="lab"><font color="Green" title="E20">A13</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E8">A1</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="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="#E12:11"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E16">A9</font></span></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></span>
;<br/>
<a NAME="E16:11"/><span class="lab"><font color="Green" title="E21">A14</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="c6">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E13:11"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="xxreal_1.html#T34" title="XXREAL_1:th.34">XXREAL_1:34</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:11"><span class="lab"><font color="Green" title="E21">A14</font></span></a>, <a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E13:11"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="xxreal_1.html#T34" title="XXREAL_1:th.34">XXREAL_1:34</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">f3</font> = <font color="Maroon" title="c9">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">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="c12">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></span>;<br/>
<span class="kw">assume </span><a NAME="E20:11"/><span class="lab"><font color="Green" title="E22">A15</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E21:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A16</font></span>: 
 <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="c12">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:11"><span class="lab"><font color="Green" title="E21">A14</font></span></a>, <a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="topmetr.html#T30" title="TOPMETR:th.30">TOPMETR:30</a></span>;<br/>
<a NAME="E22:11"/><span class="kw">then </span>
<font color="Maroon" title="c14">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:11"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="jordan6.html#T4" title="JORDAN6:th.4">JORDAN6:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c9">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c12">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="E24:11"/>
<font color="Maroon" title="c6">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:11"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E25:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A17</font></span>: 
<font color="Maroon" title="c15">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">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="E26:11"/><span class="lab"><font color="Green" title="E25">A18</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<a NAME="E27:11"/>
<font color="Maroon" title="c9">h</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="#E1:11"><span class="lab"><font color="Green" title="E8">A1</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="E28:11"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E7:11"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E8:11"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="txt" href="#E15:11"><span class="lab"><font color="Green" title="E20">A13</font></span></a>, <a class="txt" href="#E20:11"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/>
<a NAME="E29:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A19</font></span>: 
<font color="Maroon" title="c7">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">t2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E30:11"/><span class="kw">then </span>
<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#NR4" title="XXREAL_0:NR.4">&gt;</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E31:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A20</font></span>: 
<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#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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E32:11"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E33:11"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:11"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></span>;<br/>
<a NAME="E34:11"/><span class="kw">then </span>
<font color="Maroon" title="c6">s2</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="c15">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:11"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E35:11"/><span class="kw">then </span>
<font color="Maroon" title="c8">t2</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="c14">f3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:11"><span class="lab"><font color="Green" title="E24">A17</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="E36:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A21</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:11"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E37:11"/>
<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#NR4" title="XXREAL_0:NR.4">&gt;</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:11"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E38:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A22</font></span>: 
<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#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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E39:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A23</font></span>: 
<font color="Maroon" title="c3">d</font> <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:11"><span class="lab"><font color="Green" title="E28">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:11"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="txt" href="#E13:11"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="xxreal_1.html#T34" title="XXREAL_1:th.34">XXREAL_1:34</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">f4</font> = <font color="Maroon" title="c9">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">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="c11">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></span>;<br/>
<a NAME="E42:11"/><span class="lab"><font color="Green" title="E31">A24</font></span>: 
 <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="c11">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:11"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="topmetr.html#T30" title="TOPMETR:th.30">TOPMETR:30</a></span>;<br/>
<a NAME="E43:11"/><span class="kw">then </span>
<font color="Maroon" title="c18">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:11"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="jordan6.html#T4" title="JORDAN6:th.4">JORDAN6:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">f1</font> = <font color="Maroon" title="c9">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">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="E45:11"/><span class="lab"><font color="Green" title="E32">A25</font></span>: 
<font color="Maroon" title="c9">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E8">A1</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="E46:11"/><span class="kw">then </span>
<font color="Maroon" title="c18">f4</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></span>;<br/>
<a NAME="E47:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A26</font></span>: 
<font color="Maroon" title="c19">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:11"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E42:11"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="ref" href="jordan6.html#T4" title="JORDAN6:th.4">JORDAN6:4</a></span>;<br/>
<a NAME="E48:11"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E49:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A27</font></span>: 
<font color="Maroon" title="c19">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="c3">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E50:11"/>
<font color="Maroon" title="c5">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E51:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A28</font></span>: 
<font color="Maroon" title="c19">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E52:11"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E29:11"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E36:11"><span class="lab"><font color="Green" title="E28">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E54:11"/><span class="lab"><font color="Green" title="E36">A29</font></span>: 
<font color="Maroon" title="c19">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">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
 <span class="kw">and </span><br/><a NAME="E55:11"/><span class="lab"><font color="Green" title="E37">A30</font></span>: 
<font color="Maroon" title="c5">s1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">r1</font>
 <span class="kw">and </span><br/><a NAME="E56:11"/><span class="lab"><font color="Green" title="E38">A31</font></span>: 
<font color="Maroon" title="c20">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:11"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="txt" href="#E47:11"><span class="lab"><font color="Green" title="E33">A26</font></span></a>, <a class="txt" href="#E51:11"><span class="lab"><font color="Green" title="E35">A28</font></span></a>, <a class="txt" href="#E49:11"><span class="lab"><font color="Green" title="E34">A27</font></span></a>, <a class="txt" href="#E39:11"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="topreal5.html#T13" title="TOPREAL5:th.13">TOPREAL5:13</a></span>;<br/>
<a NAME="E57:11"/><span class="lab"><font color="Green" title="E39">A32</font></span>: 
<font color="Maroon" title="c20">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:11"><span class="lab"><font color="Green" title="E37">A30</font></span></a>, <a class="txt" href="#E56:11"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E58:11"/>
<font color="Maroon" title="c5">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:11"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E59:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A33</font></span>: 
<font color="Maroon" title="c15">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E60:11"/>
<font color="Maroon" title="c14">f3</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:11"><span class="lab"><font color="Green" title="E32">A25</font></span></a>, <a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></span>;<br/>
<a NAME="E61:11"/><span class="kw">then </span>
<font color="Maroon" title="c15">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:11"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E21:11"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="ref" href="jordan6.html#T4" title="JORDAN6:th.4">JORDAN6:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E63:11"/><span class="lab"><font color="Green" title="E41">A34</font></span>: 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">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
 <span class="kw">and </span><br/><a NAME="E64:11"/><span class="lab"><font color="Green" title="E42">A35</font></span>: 
<font color="Maroon" title="c6">s2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">r</font>
 <span class="kw">and </span><br/><a NAME="E65:11"/><span class="lab"><font color="Green" title="E43">A36</font></span>: 
<font color="Maroon" title="c21">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:11"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E25:11"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E59:11"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="txt" href="#E31:11"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="txt" href="#E38:11"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="ref" href="topreal5.html#T12" title="TOPREAL5:th.12">TOPREAL5:12</a></span>;<br/>
<a NAME="E66:11"/><span class="lab"><font color="Green" title="E44">A37</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="c21">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:11"><span class="lab"><font color="Green" title="E21">A14</font></span></a>, <a class="txt" href="#E64:11"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E67:11"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:11"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="txt" href="#E55:11"><span class="lab"><font color="Green" title="E37">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E68:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A38</font></span>: 
<font color="Maroon" title="c20">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:11"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E69:11"/><span class="lab"><font color="Green" title="E46">A39</font></span>: 
<font color="Maroon" title="c9">h</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="#E1:11"><span class="lab"><font color="Green" title="E8">A1</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="E70:11"/>
<font color="Maroon" title="c21">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E65:11"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E71:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A40</font></span>: 
<font color="Maroon" title="c21">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:11"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E72:11"/>
<font color="Maroon" title="c21">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:11"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="txt" href="#E65:11"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:11"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">r1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:11"><span class="lab"><font color="Green" title="E36">A29</font></span></a>, <a class="txt" href="#E57:11"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E74:11"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:11"><span class="lab"><font color="Green" title="E20">A13</font></span></a>, <a class="txt" href="#E69:11"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="txt" href="#E65:11"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="txt" href="#E71:11"><span class="lab"><font color="Green" title="E47">A40</font></span></a>, <a class="txt" href="#E55:11"><span class="lab"><font color="Green" title="E37">A30</font></span></a>, <a class="txt" href="#E68:11"><span class="lab"><font color="Green" title="E45">A38</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</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>
