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<b>let </b><font color="Maroon">P</font> be  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> 2</span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<a NAME="E2:12"/>
 <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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E3:12"/><b>then </b><i><font color="Green">E50</font></i>: 
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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>set </b><font color="Maroon">h</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">P</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>;<br/>
<b>reconsider </b><font color="Maroon">rj</font> =  <a href="pscomp_1.html#K13">proj1</a>  as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E5:12"/>
<font color="Maroon">rj</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:12"/><b>then </b>
<font color="Maroon">rj</font> <a href="tmap_1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>,<a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="tmap_1.html#T68">TMAP_1:68</a></i>;<br/>
<a NAME="E7:12"/><b>then </b>
<font color="Maroon">rj</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">P</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>,<a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#D3">TMAP_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h2</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">P</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">h2</font>
;<br/>

<b>thus </b><a NAME="E9:12"/>
<font color="Maroon">h2</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">q</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> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E10:12"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E12:12"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:12"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:12"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E15:12"/>
 <a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K13">proj1</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:12"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="pscomp_1.html#K13">proj1</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
;<br/>


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