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<b>let </b><font color="Maroon">X</font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>



<a NAME="E1:68"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" target="_self">transl</a> <font color="Maroon">a</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:68"/><i><font color="Green">E37</font></i>: 
 <a href="rltopsp1.html#K3" target="_self">transl</a> <font color="Maroon">a</font>,<font color="Maroon">X</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:68_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" target="_self">transl</a> <font color="Maroon">a</font>,<font color="Maroon">X</font></span>)</span>
 <b>and </b><br/><a NAME="E3:68_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" target="_self">transl</a> <font color="Maroon">a</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E5:68_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T4" target="_self">Th4</a>, <a class="ref" href="rltopsp1.html#T3" target="_self">Th3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rltopsp1.html#K3" target="_self">transl</a> <font color="Maroon">a</font>,<font color="Maroon">X</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rltopsp1.html#K3" target="_self">transl</a> <font color="Maroon">a</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="rltopsp1.html#E4"><i><font color="Green">Lemma44</font></i></a>, <a class="txt" href="rltopsp1.html#E5"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i><a class="txt" href="rltopsp1.html#E5"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="rltopsp1.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_2.html#T32">FUNCT_2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:68_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rltopsp1.html#K3" target="_self">transl</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rltopsp1.html#T3" target="_self">Th3</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:68"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K3" target="_self">transl</a> <font color="Maroon">a</font>,<font color="Maroon">X</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="funct_2.html#R2">=</a>  <a href="rltopsp1.html#K3" target="_self">transl</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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