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<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:30_1_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 ;<br/><b>thus </b><a NAME="E4:30_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:30_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  )
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E3:30_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:30_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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