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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">w1</font>, <font color="Olive" title="b2">w2</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 ( for <font color="Olive" title="b3">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( not <font color="Olive" title="b1">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Olive" title="b2">w2</font> &amp; not <font color="Olive" title="b2">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Olive" title="b1">w1</font> ) ) holds <br/> not  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b1">w1</font>,<font color="Olive" title="b2">w2</font></span><br/><b>let </b><font color="Maroon" title="c2">w1</font>, <font color="Maroon" title="c3">w2</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"> ( ( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( not <font color="Maroon" title="c2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">w2</font> &amp; not <font color="Maroon" title="c3">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">w1</font> ) ) implies  not  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">w1</font>,<font color="Maroon" title="c3">w2</font> )</span><br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E41">A1</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( not <font color="Maroon" title="c2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">w2</font> &amp; not <font color="Maroon" title="c3">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">w1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">w1</font>,<font color="Maroon" title="c3">w2</font></span><br/>

<b>assume </b><a NAME="E2:49"/>
 <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">w1</font>,<font color="Maroon" title="c3">w2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:49"/><b>then </b>
 <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">w1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">s</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">w2</font></span>)</span></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:49"/><i><font color="Green" title="E42">A2</font></i>: 
(  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">w1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">w2</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E6:49"/>
<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#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">w2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">w1</font>
 
<b>by </b><i><a class="txt" href="#E5:49"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>;<br/>
<a NAME="E7:49"/><b>then </b>
 <a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">w2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">w1</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self" title="JORDAN2C:th.13">Th13</a></i>;<br/>
<a NAME="E8:49"/><b>then </b><i><font color="Green" title="E43">A3</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">w1</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T44" title="EUCLID:th.44">EUCLID:44</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1"/>
(  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>suppose </b></a><a NAME="E1:49_1_1"/><i><font color="Green" title="E44">A4</font></i>: 
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:49_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">w1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:49"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><a NAME="E3:49_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">w1</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><a NAME="E4:49_1_1"/><b>then </b>
1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">w1</font>
 <b>by </b><i><a class="txt" href="#E1:49_1_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E5:49_1_1"/><b>then </b>
<font color="Maroon" title="c3">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">w1</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><b>hence </b><a NAME="E6:49_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E41">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><b>suppose </b></a><a NAME="E1:49_1_2"/>
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:49_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">w1</font>
 <b>by </b><i><a class="txt" href="#E8:49"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><b>hence </b><a NAME="E3:49_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E41">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
