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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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">R</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">q1</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">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b4">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> where <font color="Olive" title="b5">p</font> is   <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> : <font color="Olive" title="b5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> </span>}</span>  holds <br/> ex <font color="Olive" title="b5">q2</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 <br/>( <font color="Olive" title="b5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b5">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Olive" title="b1">R</font> &amp; <font color="Olive" title="b2">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b3">p2</font>,<font color="Olive" title="b4">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b5">q2</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q1</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="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> where <font color="Olive" title="b4">p</font> is   <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> : <font color="Olive" title="b4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> </span>}</span>  holds <br/> ex <font color="Olive" title="b4">q2</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 <br/>( <font color="Olive" title="b4">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b4">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b2">p2</font>,<font color="Olive" title="b3">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b4">q2</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  )</span><br/><b>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font>, <font color="Maroon" title="c5">q1</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">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> where <font color="Olive" title="b1">p</font> is   <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> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> </span>}</span>  implies  ex <font color="Olive" title="b1">q2</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 <br/>( <font color="Olive" title="b1">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">q2</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  ) )</span><br/>





<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> where <font color="Olive" title="b1">p</font> is   <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> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">q2</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 <br/>( <font color="Olive" title="b1">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">q2</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  )</span><br/>

<b>consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:60"/><i><font color="Green" title="E37">A2</font></i>: 
( <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_4.html#E60"><i><font color="Green" title="E28">Lm8</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c5">q1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E6:60"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c9">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font>
 <b>and </b><br/><a NAME="E7:60"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y2</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a> 
 <b>and </b><br/><a NAME="E8:60"/><i><font color="Green" title="E40">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <b>by </b><i><a class="txt" href="euclid_4.html#E57"><i><font color="Green" title="E27">Lm7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">q2</font> = <font color="Maroon" title="c9">y2</font> as   <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> <b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<a NAME="E10:60"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E3:60"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="#E7:60"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="euclid_2.html#D3" title="EUCLID_2:def.3">EUCLID_2:def 3</a></i>;<br/>
<a NAME="E11:60"/><b>then </b><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a> 
 
<b>by </b><i><a class="ref" href="euclid_2.html#D3" title="EUCLID_2:def.3">EUCLID_2:def 3</a></i>;<br/>
<a NAME="E12:60"/>
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 
<b>by </b><i><a class="ref" href="euclid_4.html#T47" target="_self" title="EUCLID_4:th.47">Th47</a></i>;<br/>
<a NAME="E13:60"/><b>then </b><i><font color="Green" title="E42">A7</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E14:60"/><i><font color="Green" title="E43">A8</font></i>: 
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> ex <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font> implies  ex <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">s</font> ) )</span><br/>

<b>assume </b><a NAME="E1:60_1"/><i><font color="Green" title="E44">A9</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">s</font> )</span><br/>

<b>take </b><font color="Maroon" title="c12">r</font> = <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c12">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">s</font> )</span><br/>

<b>thus </b><a NAME="E2:60_1"/>
( <font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c12">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">s</font> )
 <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="#E3:60"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="#E6:60"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="#E1:60_1"><i><font color="Green" title="E44">A9</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</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>
<a NAME="E15:60"/><i><font color="Green" title="E44">A10</font></i>: 
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> implies <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">r</font> )</span><br/>

<b>assume </b><a NAME="E1:60_2"/>
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">r</font></span><br/>

<b>then consider </b><font color="Maroon" title="c12">p0</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><b> such that </b><br/><a NAME="E3:60_2"/><i><font color="Green" title="E45">A11</font></i>: 
( <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c12">p0</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <font color="Maroon" title="c12">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:60_2"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">r</font>
 <b>by </b><i><a class="txt" href="#E3:60"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="#E8:60"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="#E3:60_2"><i><font color="Green" title="E45">A11</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>
<a NAME="E16:60"/><b>then </b>
<font color="Maroon" title="c2">R</font> is <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></i>;<br/>
<a NAME="E17:60"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c10">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="#E13:60"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="#E14:60"><i><font color="Green" title="E43">A8</font></i></a>, <a class="txt" href="#E15:60"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E18:60"/>
 ex <font color="Olive" title="b1">q2</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 <br/>( <font color="Olive" title="b1">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K4" title="EUCLID_4:func.4">Line</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">q2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c5">q1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Olive" title="b1">q2</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  )
 <b>by </b><i><a class="txt" href="#E3:60"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="#E6:60"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="#E11:60"><i><font color="Green" title="E41">A6</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>
