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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">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> <font color="Maroon" title="c1">n</font></span>)</span>  ex <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">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> st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x2</font> &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Olive" title="b3">x1</font>,<font color="Olive" title="b4">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">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> st <br/>( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font> &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c5">x2</font> = <font color="Maroon" title="c3">p2</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T22" title="EUCLID:th.22">EUCLID:22</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x2</font> &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/>

<span class="kw">thus </span><a NAME="E2:44"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span><br/>

<span class="kw">thus </span><a NAME="E3:44"/>
 <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">e</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="c4">x1</font>,<font color="Maroon" title="c5">x2</font>) or <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:44_1"/>
<font color="Maroon" title="c6">e</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="c4">x1</font>,<font color="Maroon" title="c5">x2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">lambda</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:44_1"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c6">e</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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">lambda</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">lambda</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 ;<br/>
<a NAME="E4:44_1"/>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">lambda</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">lambda</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">x1</font> &amp; <font color="Maroon" title="c7">lambda</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">lambda</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">x2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T65" title="EUCLID:th.65">EUCLID:65</a></span>;<br/>
<a NAME="E5:44_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">e</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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">lambda</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">lambda</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="euclid.html#T64" title="EUCLID:th.64">EUCLID:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:44_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) or <font color="Maroon" title="c6">e</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="c4">x1</font>,<font color="Maroon" title="c5">x2</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E4:44"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K3" title="EUCLID_4:func.3">Line</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">e</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="c4">x1</font>,<font color="Maroon" title="c5">x2</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">lambda</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:44"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c6">e</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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">lambda</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">lambda</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p2</font></span>)</span>
 ;<br/>
<a NAME="E7:44"/>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">lambda</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">lambda</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">x1</font> &amp; <font color="Maroon" title="c7">lambda</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">lambda</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">x2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T65" title="EUCLID:th.65">EUCLID:65</a></span>;<br/>
<a NAME="E8:44"/><span class="kw">then </span>
<font color="Maroon" title="c6">e</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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">lambda</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">lambda</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="euclid.html#T64" title="EUCLID:th.64">EUCLID:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:44"/>
<font color="Maroon" title="c6">e</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="c4">x1</font>,<font color="Maroon" title="c5">x2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
