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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> implies ( <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E2:40"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font>
 ; <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"> ( <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> )</span><br/>

<a NAME="E3:40"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E4:40"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="topmetr.html#T18" onmouseover="rs('topmetr/T18')" onmouseout="rh()">TOPMETR:18</a></span>;<br/>
<a NAME="E5:40"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p2">(<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="p2">(<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>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> = 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p3">(<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="p3">(<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>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p3">(<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="p3">(<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>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="treal_1.html#D1" onmouseover="rs('treal_1/D1')" onmouseout="rh()">TREAL_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="treal_1.html#T13" onmouseover="rs('treal_1/T13')" onmouseout="rh()">TREAL_1:13</a></span>
<br/>.= 
 <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">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">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">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>
;<br/> <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"> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font></span><br/>

<a NAME="E7:40"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E8:40"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="topmetr.html#T18" onmouseover="rs('topmetr/T18')" onmouseout="rh()">TOPMETR:18</a></span>;<br/>
<a NAME="E9:40"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p2">(<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="p2">(<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>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> = 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p3">(<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="p3">(<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>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p3">(<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="p3">(<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>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="treal_1.html#D2" onmouseover="rs('treal_1/D2')" onmouseout="rh()">TREAL_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="treal_1.html#T13" onmouseover="rs('treal_1/T13')" onmouseout="rh()">TREAL_1:13</a></span>
<br/>.= 
(<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</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="#E2:40"><span class="lab"><font color="Green" title="E33">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">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">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>
;<br/> <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>
