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<b>let </b><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font>, <font color="Maroon" title="c3">g1</font>, <font color="Maroon" title="c4">g2</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</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="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c4">g2</font> <a href="jordan12.html#R2" title="JORDAN12:pred.2">are_in_general_position</a>  implies <font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">g1</font> <a href="jordan12.html#R2" title="JORDAN12:pred.2">are_in_general_position</a>  )</span><br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c4">g2</font> <a href="jordan12.html#R2" title="JORDAN12:pred.2">are_in_general_position</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">g1</font> <a href="jordan12.html#R2" title="JORDAN12:pred.2">are_in_general_position</a> </span><br/>

<a NAME="E2:13"/><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font> <a href="jordan12.html#R1" title="JORDAN12:pred.1">is_in_general_position_wrt</a> <font color="Maroon" title="c3">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c4">g2</font> &amp; <font color="Maroon" title="c3">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c4">g2</font> <a href="jordan12.html#R1" title="JORDAN12:pred.1">is_in_general_position_wrt</a> <font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E1:13"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="jordan12.html#D2" target="_self" title="JORDAN12:def.2">Def2</a></i>;<br/>
<a NAME="E3:13"/><b>then </b>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan12.html#D1" target="_self" title="JORDAN12:def.1">Def1</a></i>;<br/>
<a NAME="E4:13"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">g1</font>
 
<b>by </b><i><a class="ref" href="topreal8.html#T10" title="TOPREAL8:th.10">TOPREAL8:10</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E5:13"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:13"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="jordan12.html#D1" target="_self" title="JORDAN12:def.1">Def1</a></i>;<br/>
<a NAME="E6:13"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c3">g1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan12.html#T6" target="_self" title="JORDAN12:th.6">Th6</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<div><i><font color="Green" title="E16">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span> implies <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )</span><br/><b>assume </b><a NAME="E1:13_1"/><i><font color="Green" title="E17">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a></span><br/><a NAME="E2:13_1"/><i><font color="Green" title="E18">A7</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c3">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan12.html#T6" target="_self" title="JORDAN12:th.6">Th6</a></i>;<br/><a NAME="E3:13_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c4">g2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:13"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="jordan12.html#E1:13_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="jordan12.html#D1" target="_self" title="JORDAN12:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:13_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:13_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="sprect_1.html#T1" title="SPRECT_1:th.1">SPRECT_1:1</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</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="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> implies <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c5">i</font></span>)</span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )</span><br/><b>assume </b><a NAME="E1:13_2"/><i><font color="Green" title="E17">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c5">i</font></span>)</span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a></span><br/><a NAME="E2:13_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal8.html#T7" title="TOPREAL8:th.7">TOPREAL8:7</a></i>;<br/><a NAME="E3:13_2"/><b>then </b>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:13_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:13_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c4">g2</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:13"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="jordan12.html#E1:13_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="jordan12.html#D1" target="_self" title="JORDAN12:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E5:13_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c5">i</font></span>)</span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="jordan12.html#E1:13_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="topreal8.html#T28" title="TOPREAL8:th.28">TOPREAL8:28</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="E9:13"/><b>then </b>
( <font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font> <a href="jordan12.html#R1" title="JORDAN12:pred.1">is_in_general_position_wrt</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c3">g1</font> <a href="jordan12.html#R1" title="JORDAN12:pred.1">is_in_general_position_wrt</a> <font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font> )
 
<b>by </b><i><a class="txt" href="jordan12.html#E4:13"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="jordan12.html#E6:13"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="jordan12.html#E7:13"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="jordan12.html#D1" target="_self" title="JORDAN12:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E10:13"/>
<font color="Maroon" title="c1">f1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">g1</font> <a href="jordan12.html#R2" title="JORDAN12:pred.2">are_in_general_position</a> 
 <b>by </b><i><a class="ref" href="jordan12.html#D2" target="_self" title="JORDAN12:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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