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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Olive" title="b3">z</font> holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font></span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">z</font> implies <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> )</span><br/>







<a NAME="E1:2"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></i>;<br/>
<b>assume </b><a NAME="E2:2"/>
<font color="Maroon" title="c2">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font></span><br/>

<a NAME="E3:2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T9" title="JORDAN2C:th.9">JORDAN2C:9</a></i>;<br/>
<a NAME="E4:2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">z</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T9" title="JORDAN2C:th.9">JORDAN2C:9</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c3">y</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">z</font>
<b>by </b><i><a class="txt" href="topreal9.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon" title="c4">z</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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