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<b>let </b><font color="Maroon">N</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">w1</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">N</font></span>)</span>;<br/><b>let </b><font color="Maroon">w2</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">N</font></span>)</span>;<br/><b>let </b><font color="Maroon">w</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">N</font></span>)</span>;<br/>







 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">N</font> = 
<font color="Maroon">w</font> <a href="euclid.html#K20">-</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
;<br/>
<b>then </b><span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">w2</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> = 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">w2</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">w2</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">w2</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="euclid.html#K20">-</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>
;<br/>
<b>hence </b><a NAME="E3:41"/>
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">w2</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">w</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="euclid.html#K20">-</a> <font color="Maroon">w2</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 <b>by </b><i/>;<br/>


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