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<b>let </b><font color="Maroon" title="c7">POS</font> be   <a href="analmetr.html#NM1" title="ANALMETR:NM.1">OrtAfSp</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">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">POS</font><br/> for <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span>  st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y</font> holds <br/>( <font color="Olive" title="b1">X</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> iff <font color="Olive" title="b2">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )</span><br/><b>let </b><font color="Maroon" title="c8">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">POS</font>; <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">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span>  st <font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> holds <br/>( <font color="Maroon" title="c8">X</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> iff <font color="Olive" title="b1">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )</span><br/><b>let </b><font color="Maroon" title="c9">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</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="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Y</font> implies ( <font color="Maroon" title="c8">X</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> iff <font color="Maroon" title="c9">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> ) )</span><br/>





<b>assume </b><a NAME="E1:70"/><i><font color="Green" title="E55">A1</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">X</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> iff <font color="Maroon" title="c9">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> implies <font color="Maroon" title="c8">X</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:70_1"/>
<font color="Maroon" title="c8">X</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a></span><br/><b>then consider </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font><b> such that </b><br/><a NAME="E3:70_1"/><i><font color="Green" title="E56">A2</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b</font> &amp; <font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#D13" target="_self" title="ANALMETR:def.13">Def13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">a'</font> = <font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c13">b'</font> = <font color="Maroon" title="c11">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> ;<br/><a NAME="E5:70_1"/>
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font>
 <b>by </b><i><a class="txt" href="#E1:70"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="#E3:70_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="analmetr.html#T56" target="_self" title="ANALMETR:th.56">Th56</a></i>;<br/><b>hence </b><a NAME="E6:70_1"/>
<font color="Maroon" title="c9">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 <b>by </b><i><a class="txt" href="#E3:70_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</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>

<b>assume </b><a NAME="E3:70"/>
<font color="Maroon" title="c9">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">X</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a></span><br/>

<b>then consider </b><font color="Maroon" title="c10">a'</font>, <font color="Maroon" title="c11">b'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span><b> such that </b><br/><a NAME="E5:70"/><i><font color="Green" title="E56">A3</font></i>: 
( <font color="Maroon" title="c10">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b'</font> &amp; <font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c11">b'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c10">a'</font>, <font color="Maroon" title="c13">b</font> = <font color="Maroon" title="c11">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> ;<br/>
<a NAME="E7:70"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font>
 
<b>by </b><i><a class="txt" href="#E1:70"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="#E5:70"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="analmetr.html#T56" target="_self" title="ANALMETR:th.56">Th56</a></i>;<br/>
<b>hence </b><a NAME="E8:70"/>
<font color="Maroon" title="c8">X</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>
 <b>by </b><i><a class="txt" href="#E5:70"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="analmetr.html#D13" target="_self" title="ANALMETR:def.13">Def13</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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