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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</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">q1</font>, <font color="Olive" title="b2">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><br/>  for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">i</font>) &amp; <font color="Olive" title="b2">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">i</font>) &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font>, <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> holds <br/> <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font>, <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b3">i</font>),<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">q1</font>, <font color="Maroon" title="c3">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">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c2">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon" title="c3">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>, <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> holds <br/> <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>, <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font>),<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>) &amp; <font color="Maroon" title="c3">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>) &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>, <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> implies  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>, <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>),<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c2">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">and </span><br/><a NAME="E2:36"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c3">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">and </span><br/><a NAME="E3:36"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> 
 <span class="kw">and </span><br/><a NAME="E4:36"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</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"> (  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>, <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>, <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>),<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )</span><br/>

<a NAME="E5:36"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="topreal1.html#D8" onmouseover="rs('topreal1/D8')" onmouseout="rh()">TOPREAL1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">P</font> =  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c6">Q</font> =  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>) 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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="topreal1.html#T23" onmouseover="rs('topreal1/T23')" onmouseout="rh()">TOPREAL1:23</a></span>;<br/>
<a NAME="E7:36"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="topreal1.html#T25" onmouseover="rs('topreal1/T25')" onmouseout="rh()">TOPREAL1:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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="c5">P</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:36"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
( <font color="Maroon" title="c7">F</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</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="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &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="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D1" onmouseover="rs('topreal1/D1')" onmouseout="rh()">TOPREAL1:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">ppi</font>, <font color="Maroon" title="c9">pi1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:36"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c8">ppi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">pi1</font>
 <span class="kw">and </span><br/><a NAME="E12:36"/><span class="lab"><font color="Green" title="E33">A7</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="c8">ppi</font>
 <span class="kw">and </span><br/><a NAME="E13:36"/>
<font color="Maroon" title="c8">ppi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E14:36"/>
 <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">pi1</font>
 <span class="kw">and </span><br/><a NAME="E15:36"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<font color="Maroon" title="c9">pi1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E16:36"/><span class="lab"><font color="Green" title="E35">A9</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E17:36"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">ppi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E18:36"/><span class="lab"><font color="Green" title="E37">A11</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">pi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="jordan5b.html#T7" onmouseover="rs('jordan5b/T7')" onmouseout="rh()">JORDAN5B:7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">Ex</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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p1">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>);<br/>
<a NAME="E19:36"/><span class="lab"><font color="Green" title="E38">A12</font></span>: 
 <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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p1">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>) is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="treal_1.html#T17" onmouseover="rs('treal_1/T17')" onmouseout="rh()">TREAL_1:17</a></span>;<br/>
<a NAME="E20:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A13</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>;<br/>
<a NAME="E21:36"/><span class="lab"><font color="Green" title="E40">A14</font></span>: 
<font color="Maroon" title="c8">ppi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">dd</font> where <font color="Olive" title="b1">dd</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c8">ppi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">dd</font> &amp; <font color="Olive" title="b1">dd</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">pi1</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E32">A6</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">Poz</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a>, <a class="ref" href="xxreal_1.html#T34" onmouseover="rs('xxreal_1/T34')" onmouseout="rh()">XXREAL_1:34</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">G</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#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Poz</font></span>)</span>,<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="c6">Q</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E24:36"/><span class="lab"><font color="Green" title="E41">A15</font></span>: 
<font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">Poz</font>
 <span class="kw">and </span><br/><a NAME="E25:36"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
<font color="Maroon" title="c12">G</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E16:36"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="jordan5b.html#T8" onmouseover="rs('jordan5b/T8')" onmouseout="rh()">JORDAN5B:8</a></span>;<br/>
<a NAME="E26:36"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
<font color="Maroon" title="c8">ppi</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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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="#E21:36"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <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">set </span><font color="Maroon" title="c13">H</font> = <font color="Maroon" title="c12">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>)</span>)</span>;<br/>
<span class="lab"><font color="Green" title="E44">A18</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">G</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Poz</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:36"><span class="lab"><font color="Green" title="E42">A16</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="c11">Poz</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 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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="topmetr.html#T18" onmouseover="rs('topmetr/T18')" onmouseout="rh()">TOPMETR:18</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E45">A19</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p3">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>)</span>)</span></span>)</span> = 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="ref" href="relat_1.html#T28" onmouseover="rs('relat_1/T28')" onmouseout="rh()">RELAT_1:28</a></span>
<br/>.= 
 <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> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:36"><span class="lab"><font color="Green" title="E42">A16</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/>
<a NAME="E29:36"/>
<font color="Maroon" title="c9">pi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">dd</font> where <font color="Olive" title="b1">dd</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c8">ppi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">dd</font> &amp; <font color="Olive" title="b1">dd</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">pi1</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E32">A6</font></span></a></span>;<br/>
<a NAME="E30:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A20</font></span>: 
<font color="Maroon" title="c9">pi1</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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E47">A21</font></span>: <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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> 1 = 
<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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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="ref" href="treal_1.html#D2" onmouseover="rs('treal_1/D2')" onmouseout="rh()">TREAL_1:def 2</a></span>
<br/>.= 
(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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="#E11:36"><span class="lab"><font color="Green" title="E32">A6</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="c9">pi1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E32">A6</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="lab"><font color="Green" title="E48">A22</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p3">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>)</span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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="#E20:36"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E27:36"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>
<br/>.= 
 <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="#E19:36"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</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="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">H</font> = <font color="Maroon" title="c12">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>)</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>,<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="c6">Q</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:36"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/>
<a NAME="E34:36"/>
<font color="Maroon" title="c2">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="c14">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E28:36"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">x19</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E36:36"/><span class="lab"><font color="Green" title="E49">A23</font></span>: 
<font color="Maroon" title="c15">x19</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="c14">H</font>
 <span class="kw">and </span><br/><a NAME="E37:36"/><span class="lab"><font color="Green" title="E50">A24</font></span>: 
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x19</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="E38:36"/>
<font color="Maroon" title="c15">x19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</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">l</font> &amp; <font color="Olive" title="b1">l</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="txt" href="#E32:36"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E36:36"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <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="c16">x1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E40:36"/><span class="lab"><font color="Green" title="E51">A25</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x19</font>
 <span class="kw">and </span><br/><a NAME="E41:36"/><span class="lab"><font color="Green" title="E52">A26</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="c16">x1</font> &amp; <font color="Maroon" title="c16">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>
<span class="kw">assume </span><a NAME="E42:36"/><span class="lab"><font color="Green" title="E53">A27</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>, <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></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">  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>, <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>),<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/>

<a NAME="E43:36"/>
<font color="Maroon" title="c16">x1</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:36"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<a NAME="E44:36"/><span class="kw">then </span>
<font color="Maroon" title="c16">x1</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#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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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="#E20:36"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E27:36"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E32:36"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E45:36"/><span class="kw">then </span>
<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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> <font color="Maroon" title="c16">x1</font> <a href="tarski.html#R2" title="TARSKI: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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E46:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A28</font></span>: 
<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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> <font color="Maroon" title="c16">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">Poz</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="topmetr.html#T18" onmouseover="rs('topmetr/T18')" onmouseout="rh()">TOPMETR:18</a></span>;<br/>
<a NAME="E47:36"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:36"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E55">A29</font></span>: <font color="Maroon" title="c14">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c12">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">pi1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:36"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:36"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="txt" href="#E30:36"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="txt" href="#E24:36"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="lab"><font color="Green" title="E56">A30</font></span>: <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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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="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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E32">A6</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="c8">ppi</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E32">A6</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 NAME="E50:36"/>
<font color="Maroon" title="c3">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="c14">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E28:36"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x29</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E52:36"/><span class="lab"><font color="Green" title="E57">A31</font></span>: 
<font color="Maroon" title="c17">x29</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="c14">H</font>
 <span class="kw">and </span><br/><a NAME="E53:36"/><span class="lab"><font color="Green" title="E58">A32</font></span>: 
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x29</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="E54:36"/>
<font color="Maroon" title="c17">x29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</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">l</font> &amp; <font color="Olive" title="b1">l</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="txt" href="#E32:36"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E52:36"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <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="c18">x2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E56:36"/><span class="lab"><font color="Green" title="E59">A33</font></span>: 
<font color="Maroon" title="c18">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x29</font>
 <span class="kw">and </span><br/><a NAME="E57:36"/><span class="lab"><font color="Green" title="E60">A34</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="c18">x2</font>
 <span class="kw">and </span><br/><a NAME="E58:36"/><span class="lab"><font color="Green" title="E61">A35</font></span>: 
<font color="Maroon" title="c18">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">X1</font> = <font color="Maroon" title="c16">x1</font>, <font color="Maroon" title="c20">X2</font> = <font color="Maroon" title="c18">x2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</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="#E41:36"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="txt" href="#E57:36"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E58:36"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/>
<a NAME="E60:36"/>
<font color="Maroon" title="c18">x2</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:36"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E58:36"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<a NAME="E61:36"/><span class="kw">then </span>
<font color="Maroon" title="c18">x2</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#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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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="#E20:36"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E27:36"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E32:36"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E62:36"/><span class="kw">then </span>
<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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> <font color="Maroon" title="c18">x2</font> <a href="tarski.html#R2" title="TARSKI: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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E63:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A36</font></span>: 
<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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> <font color="Maroon" title="c18">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">Poz</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="topmetr.html#T18" onmouseover="rs('topmetr/T18')" onmouseout="rh()">TOPMETR:18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">E1</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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> <font color="Maroon" title="c19">X1</font>, <font color="Maroon" title="c22">E2</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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p2">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> <font color="Maroon" title="c20">X2</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:36"><span class="lab"><font color="Green" title="E54">A28</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E63">A37</font></span>: <font color="Maroon" title="c3">q2</font> = 
<font color="Maroon" title="c12">G</font> <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#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p3">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> <font color="Maroon" title="c18">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:36"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="txt" href="#E53:36"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="txt" href="#E56:36"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<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"><span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p3">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> <font color="Maroon" title="c18">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:36"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E63:36"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">K1</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>), <font color="Maroon" title="c24">K2</font> = <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Poz</font> as    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a>, <a class="ref" href="treal_1.html#T3" onmouseover="rs('treal_1/T3')" onmouseout="rh()">TREAL_1:3</a></span>;<br/>
<a NAME="E67:36"/><span class="lab"><font color="Green" title="E64">A38</font></span>: 
(  <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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p1">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>) <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a></span>)</span>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p1">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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="#E19:36"><span class="lab"><font color="Green" title="E38">A12</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 <font color="Maroon" title="c23">K1</font> = 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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="#E11:36"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="topmetr.html#T18" onmouseover="rs('topmetr/T18')" onmouseout="rh()">TOPMETR:18</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Poz</font></span>)</span>
<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/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c24">K2</font>

;<br/>
<a NAME="E69:36"/><span class="kw">then </span>
<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Poz</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="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T5" onmouseover="rs('tsep_1/T5')" onmouseout="rh()">TSEP_1:5</a></span>;<br/>
<a NAME="E70:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A39</font></span>: 
<font color="Maroon" title="c14">H</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:36"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E19:36"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a>, <a class="ref" href="tops_2.html#T57" onmouseover="rs('tops_2/T57')" onmouseout="rh()">TOPS_2:57</a></span>;<br/>
<span class="lab"><font color="Green" title="E66">A40</font></span>: <font color="Maroon" title="c2">q1</font> = 
<font color="Maroon" title="c12">G</font> <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#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p3">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> <font color="Maroon" title="c16">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:36"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="txt" href="#E37:36"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E40:36"><span class="lab"><font color="Green" title="E51">A25</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<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"><span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> (<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</font>)</span>)</span>,<span class="p3">(<span class="default">(<font color="Maroon" title="c8">ppi</font>,<font color="Maroon" title="c9">pi1</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> <font color="Maroon" title="c16">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:36"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E46:36"><span class="lab"><font color="Green" title="E54">A28</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<a NAME="E72:36"/><span class="lab"><font color="Green" title="E67">A41</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="c22">E2</font> &amp; <font color="Maroon" title="c22">E2</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="#E63:36"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<a NAME="E73:36"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:36"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E68">A42</font></span>: <font color="Maroon" title="c14">H</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="c12">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">ppi</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:36"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:36"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E26:36"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E24:36"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<a NAME="E75:36"/>
<font color="Maroon" title="c21">E1</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="#E46:36"><span class="lab"><font color="Green" title="E54">A28</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<a NAME="E76:36"/><span class="kw">then </span>
<font color="Maroon" title="c21">E1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">E2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:36"><span class="lab"><font color="Green" title="E53">A27</font></span></a>, <a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E71:36"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E65:36"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="txt" href="#E72:36"><span class="lab"><font color="Green" title="E67">A41</font></span></a></span>;<br/>
<a NAME="E77:36"/><span class="kw">then </span>
<font color="Maroon" title="c16">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E49:36"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="txt" href="#E31:36"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E67:36"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="ref" href="jordan5a.html#T15" onmouseover="rs('jordan5a/T15')" onmouseout="rh()">JORDAN5A:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E78:36"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>, <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font>),<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E70:36"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="txt" href="#E74:36"><span class="lab"><font color="Green" title="E68">A42</font></span></a>, <a class="txt" href="#E48:36"><span class="lab"><font color="Green" title="E55">A29</font></span></a>, <a class="txt" href="#E37:36"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E53:36"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="txt" href="#E40:36"><span class="lab"><font color="Green" title="E51">A25</font></span></a>, <a class="txt" href="#E41:36"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="txt" href="#E56:36"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="txt" href="#E58:36"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="ref" href="jordan5c.html#T8" target="_self" onmouseover="rs('jordan5c/T8')" onmouseout="rh()">Th8</a>, <a class="ref" href="jordan5b.html#T15" onmouseover="rs('jordan5b/T15')" onmouseout="rh()">JORDAN5B:15</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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