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<span class="kw">let </span><font color="Maroon" title="c1">x3</font>, <font color="Maroon" title="c2">y3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Maroon" title="c1">x3</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>,<span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Olive" title="b3">y1</font>,<font color="Olive" title="b4">y2</font>,<font color="Maroon" title="c2">y3</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b3">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b4">y2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">y3</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font>, <font color="Maroon" title="c5">y1</font>, <font color="Maroon" title="c6">y2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c1">x3</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>,<span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">y3</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">y2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">y3</font></span>)</span></span><br/>











<span class="kw">consider </span><font color="Maroon" title="c7">p1</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> 3</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c1">x3</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 ;<br/>
<a NAME="E3:31"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c7">p1</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">p2</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> 3</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:31"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c8">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">y3</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 ;<br/>
<a NAME="E6:31"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c8">p2</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<a NAME="E7:31"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
( <font color="Maroon" title="c8">p2</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c8">p2</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<a NAME="E8:31"/>
( <font color="Maroon" title="c7">p1</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font> &amp; <font color="Maroon" title="c7">p1</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:31"/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c1">x3</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>,<span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">y3</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">y2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">y3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="euclid_5.html#T29" target="_self" title="EUCLID_5:th.29">Th29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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