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<span class="kw">let </span><font color="Maroon" title="c2">T</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">T</font> is  <a href="topalg_2.html#V2" title="TOPALG_2:attr.2">interval</a>  implies <font color="Maroon" title="c2">T</font> is  <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">pathwise_connected</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c2">T</font> is  <a href="topalg_2.html#V2" title="TOPALG_2:attr.2">interval</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">T</font> is  <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">pathwise_connected</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="borsuk_2.html#D3" onmouseover="rs('borsuk_2/D3')" onmouseout="rh()">BORSUK_2:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:23_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_1_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span>,<span class="p1">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>);<br/><span class="kw">set </span><font color="Maroon" title="c6">X</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>);<br/><a NAME="E2:23_1_1_1_1"/><span class="lab"><font color="Green" title="E19">A3</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">a</font>,<font color="Maroon" title="c4">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="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="topmetr.html#T18" onmouseover="rs('topmetr/T18')" onmouseout="rh()">TOPMETR:18</a></span>;<br/><a NAME="E3:23_1_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="topalg_2.html#T8" target="_self" onmouseover="rs('topalg_2/T8')" onmouseout="rh()">Th8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">f</font> =  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span>,<span class="p1">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>) as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="funct_2.html#T7" onmouseover="rs('funct_2/T7')" onmouseout="rh()">FUNCT_2:7</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c7">f</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="borsuk_2.html#D1" onmouseover="rs('borsuk_2/D1')" onmouseout="rh()">BORSUK_2:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )</span><br/><a NAME="E5:23_1_1_1_1"/>
 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">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" onmouseover="rs('tsep_1/T1')" onmouseout="rh()">TSEP_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" onmouseover="rs('funct_2/T7')" onmouseout="rh()">FUNCT_2:7</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/><a NAME="E7:23_1_1_1_1"/>
 <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span>,<span class="p1">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>) 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="#E1:23_1_1_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="treal_1.html#T8" onmouseover="rs('treal_1/T8')" onmouseout="rh()">TREAL_1:8</a></span>;<br/><a NAME="E8:23_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">g</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="pre_topc.html#T26" onmouseover="rs('pre_topc/T26')" onmouseout="rh()">PRE_TOPC:26</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/><span class="kw">hence </span><a NAME="E9:23_1_1_1_1"/>
<font color="Maroon" title="c7">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="ref" href="pre_topc.html#T27" onmouseover="rs('pre_topc/T27')" onmouseout="rh()">PRE_TOPC:27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#D1" onmouseover="rs('treal_1/D1')" onmouseout="rh()">TREAL_1:def 1</a></span>
<br/>.= 
 <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="treal_1.html#T9" onmouseover="rs('treal_1/T9')" onmouseout="rh()">TREAL_1:9</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="treal_1.html#D1" onmouseover="rs('treal_1/D1')" onmouseout="rh()">TREAL_1:def 1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#D2" onmouseover="rs('treal_1/D2')" onmouseout="rh()">TREAL_1:def 2</a></span>
<br/>.= 
(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="treal_1.html#T9" onmouseover="rs('treal_1/T9')" onmouseout="rh()">TREAL_1:9</a></span>
<br/>.= 
<font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="treal_1.html#D2" onmouseover="rs('treal_1/D2')" onmouseout="rh()">TREAL_1:def 2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_1_2"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span>);<br/><span class="kw">set </span><font color="Maroon" title="c6">X</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>);<br/><a NAME="E2:23_1_1_1_2"/><span class="lab"><font color="Green" title="E19">A5</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="c4">b</font>,<font color="Maroon" title="c3">a</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="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1_2"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="topmetr.html#T18" onmouseover="rs('topmetr/T18')" onmouseout="rh()">TOPMETR:18</a></span>;<br/><a NAME="E3:23_1_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="topalg_2.html#T8" target="_self" onmouseover="rs('topalg_2/T8')" onmouseout="rh()">Th8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">f</font> =  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span>) as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1_1_2"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="funct_2.html#T7" onmouseover="rs('funct_2/T7')" onmouseout="rh()">FUNCT_2:7</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c7">f</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="borsuk_2.html#D1" onmouseover="rs('borsuk_2/D1')" onmouseout="rh()">BORSUK_2:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )</span><br/><a NAME="E5:23_1_1_1_2"/>
 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="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" onmouseover="rs('tsep_1/T1')" onmouseout="rh()">TSEP_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" onmouseover="rs('funct_2/T7')" onmouseout="rh()">FUNCT_2:7</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/><a NAME="E7:23_1_1_1_2"/>
 <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span>) 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="#E1:23_1_1_1_2"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="treal_1.html#T8" onmouseover="rs('treal_1/T8')" onmouseout="rh()">TREAL_1:8</a></span>;<br/><a NAME="E8:23_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">g</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="pre_topc.html#T26" onmouseover="rs('pre_topc/T26')" onmouseout="rh()">PRE_TOPC:26</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/><span class="kw">hence </span><a NAME="E9:23_1_1_1_2"/>
<font color="Maroon" title="c7">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="ref" href="pre_topc.html#T27" onmouseover="rs('pre_topc/T27')" onmouseout="rh()">PRE_TOPC:27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#D1" onmouseover="rs('treal_1/D1')" onmouseout="rh()">TREAL_1:def 1</a></span>
<br/>.= 
(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1_2"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="treal_1.html#T9" onmouseover="rs('treal_1/T9')" onmouseout="rh()">TREAL_1:9</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1_2"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="treal_1.html#D2" onmouseover="rs('treal_1/D2')" onmouseout="rh()">TREAL_1:def 2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#D2" onmouseover="rs('treal_1/D2')" onmouseout="rh()">TREAL_1:def 2</a></span>
<br/>.= 
 <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1_2"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="treal_1.html#T9" onmouseover="rs('treal_1/T9')" onmouseout="rh()">TREAL_1:9</a></span>
<br/>.= 
<font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_1_2"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="treal_1.html#D1" onmouseover="rs('treal_1/D1')" onmouseout="rh()">TREAL_1:def 1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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