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<span class="kw">let </span><font color="Maroon" title="c1">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</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">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b4">q2</font>,<font color="Olive" title="b3">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> holds <br/><font color="Olive" title="b3">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</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="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> implies <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:15"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E3:15"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</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="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:15"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E6:15"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
( <font color="Maroon" title="c6">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="c2">p1</font> &amp; <font color="Maroon" title="c6">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="c3">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="topreal1.html#D1" onmouseover="rs('topreal1/D1')" onmouseout="rh()">TOPREAL1:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E17">A6</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>

;<br/>
<span class="lab"><font color="Green" title="E18">A7</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E9:15"/><span class="kw">then </span>
<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E11:15"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">and </span><br/><a NAME="E12:15"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E13:15"/>
<font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E18">A7</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E15:15"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c8">y</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="c6">f</font>
 <span class="kw">and </span><br/><a NAME="E16:15"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E17:15"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r1</font> where <font color="Olive" title="b1">r1</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">s3</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E19:15"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<font color="Maroon" title="c9">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E20:15"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s3</font> &amp; <font color="Maroon" title="c9">s3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a></span>;<br/>
<a NAME="E21:15"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r1</font> where <font color="Olive" title="b1">r1</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">s4</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E23:15"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
<font color="Maroon" title="c10">s4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">and </span><br/><a NAME="E24:15"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s4</font> &amp; <font color="Maroon" title="c10">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E15:15"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a></span>;<br/>
<a NAME="E25:15"/>
( <font color="Maroon" title="c9">s3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s4</font> &amp; <font color="Maroon" title="c10">s4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E12:15"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E19:15"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E20:15"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="txt" href="#E24:15"><span class="lab"><font color="Green" title="E26">A15</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:15"/>
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:15"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E19:15"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</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/>


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