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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>given </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E1:31"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tmap_1.html#NK5" target="_self">incl</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E4:31"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tmap_1.html#NK5" target="_self">incl</a> <a href="topmetr.html#K3">R^1</a>  is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="tmap_1.html#T98">TMAP_1:98</a></i>;<br/>
<a NAME="E5:31"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tmap_1.html#NK5" target="_self">incl</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green">E28</font></i>: 
 <a href="topmetr.html#K3">R^1</a>  is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/>
<a NAME="E7:31"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<a NAME="E8:31"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="txt" href="treal_1.html#E2:31"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:31"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:31"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tmap_1.html#NK5" target="_self">incl</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tmap_1.html#NK5" target="_self">incl</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/>
<a NAME="E12:31"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tmap_1.html#NK5" target="_self">incl</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="tmap_1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#D6">TMAP_1:def 6</a></i>;<br/>
<a NAME="E13:31"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tmap_1.html#NK5" target="_self">incl</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T61">TMAP_1:61</a></i>;<br/>
<a NAME="E14:31"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tmap_1.html#NK5" target="_self">incl</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>;<br/>
<a NAME="E15:31"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tmap_1.html#NK5" target="_self">incl</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="treal_1.html#E3:31"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T162">FUNCT_1:162</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:31_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tmap_1.html#NK5" target="_self">incl</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="treal_1.html#E2:31"><i><font color="Green">E25</font></i></a>, <a class="txt" href="treal_1.html#E5:31"><i><font color="Green">E27</font></i></a>, <a class="txt" href="treal_1.html#E6:31"><i><font color="Green">E28</font></i></a>, <a class="txt" href="treal_1.html#E8:31"><i><font color="Green">E29</font></i></a>, <a class="txt" href="treal_1.html#E15:31"><i><font color="Green">E30</font></i></a>, <a class="ref" href="treal_1.html#T26" target="_self">Th26</a></i>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E3:31_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tmap_1.html#NK5" target="_self">incl</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="tmap_1.html#T95">TMAP_1:95</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="treal_1.html#E2:31_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:31"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


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