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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>





<span class="kw">assume </span><a NAME="E1:85"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">assume </span><a NAME="E1:85_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:85_1"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:85_1"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D13" target="_self" title="ANALMETR:def.13">Def13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">a9</font> = <font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c13">b9</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="E6:85_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">a9</font>,<font color="Maroon" title="c13">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E4:85_1"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="analmetr.html#T56" target="_self" title="ANALMETR:th.56">Th56</a></span>;<br/><span class="kw">hence </span><a NAME="E7:85_1"/>
<font color="Maroon" title="c9">Y</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85_1"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:85"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">a9</font>, <font color="Maroon" title="c11">b9</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><span class="kw"> such that </span><br/><a NAME="E5:85"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
<font color="Maroon" title="c10">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b9</font>
 <span class="kw">and </span><br/><a NAME="E6:85"/><span class="lab"><font color="Green" title="E57">A5</font></span>: 
<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">a9</font>,<font color="Maroon" title="c11">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c10">a9</font>, <font color="Maroon" title="c13">b</font> = <font color="Maroon" title="c11">b9</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="E8:85"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E6:85"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="analmetr.html#T56" target="_self" title="ANALMETR:th.56">Th56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:85"/>
<font color="Maroon" title="c8">X</font> is  <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:85"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="analmetr.html#D13" target="_self" title="ANALMETR:def.13">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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