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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p1</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>, <font color="Maroon">p2</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>, <font color="Maroon">p3</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>, <font color="Maroon">p</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>assume </b><a NAME="E1:56"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K9" target="_self">plane</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K8" target="_self">outside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K6" target="_self">closed_inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1"><b>case </b></a><a NAME="E1:56_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K8" target="_self">outside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p4</font> being   <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><b> such that </b><br/><a NAME="E3:56_1_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( ( 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">a1</font> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">a2</font> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">a3</font> ) &amp; <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) )
 ;<br/><b>consider </b><font color="Maroon">a01</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a02</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a03</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:56_1_1_1"/><i><font color="Green">E50</font></i>: 
( ( 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a01</font> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a02</font> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a03</font> ) &amp; <span class="p1">(<span class="default"><font color="Maroon">a01</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a02</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a03</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a01</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">a02</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">a03</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:56_1_1_1"/>
 ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2"><b>case </b></a><a NAME="E1:56_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_3.html#K6" target="_self">closed_inside_of_triangle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p4</font> being   <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><b> such that </b><br/><a NAME="E3:56_1_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a3</font> &amp; <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) )
 ;<br/><b>consider </b><font color="Maroon">a01</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a02</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a03</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:56_1_1_2"/><i><font color="Green">E71</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a01</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a02</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a03</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a01</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a02</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a03</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a01</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">a02</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">a03</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:56_1_1_2"/>
 ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:56"/>
 ex <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> )
 ;<br/>


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