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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:51"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:51"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i><a class="ref" href="jordan2c.html#T44" target="_self">Th44</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E1:51"><i><font color="Green">E45</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><b>suppose </b></a><a NAME="E1:51_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:51_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:51_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:51_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:51_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:51_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:51_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E7:51_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:51"><i><font color="Green">E45</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E8:51_1_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:51_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E9:51_1_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E10:51_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E11:51_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="topmetr.html#T4">TOPMETR:4</a></i>;<br/><a NAME="E12:51_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="jordan2c.html#E7:51_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E14:51_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><a NAME="E16:51_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:51_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan2c.html#E11:51_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/><a NAME="E17:51_1_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E1:51"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E19:51_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:51"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E20:51_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:51_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan2c.html#E16:51_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan2c.html#E41"><i><font color="Green">Lemma32</font></i></a></i>;<br/><b>hence </b><a NAME="E21:51_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_2"><b>suppose </b></a><a NAME="E1:51_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:51_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:51"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
