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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">K</font>, <font color="Olive" title="b3">M</font>, <font color="Olive" title="b4">N</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="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font>  st <font color="Olive" title="b7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">M</font> &amp; <font color="Olive" title="b7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">N</font> &amp; <font color="Olive" title="b5">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">M</font> &amp; <font color="Olive" title="b6">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">N</font> &amp; <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b</font> &amp; <font color="Olive" title="b5">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">K</font> &amp; <font color="Olive" title="b6">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">K</font> &amp; <font color="Olive" title="b1">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Olive" title="b3">M</font> &amp; <font color="Olive" title="b1">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Olive" title="b4">N</font> &amp; <font color="Olive" title="b2">K</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  holds <br/><font color="Olive" title="b1">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Olive" title="b2">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">A</font>, <font color="Maroon" title="c9">K</font>, <font color="Maroon" title="c10">M</font>, <font color="Maroon" title="c11">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">POS</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font>  st <font color="Olive" title="b3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Olive" title="b3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K</font> &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K</font> &amp; <font color="Maroon" title="c8">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c8">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c9">K</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  holds <br/><font color="Maroon" title="c8">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c9">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font>, <font color="Maroon" title="c14">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c14">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c12">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c13">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c12">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K</font> &amp; <font color="Maroon" title="c13">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K</font> &amp; <font color="Maroon" title="c8">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c8">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c9">K</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  implies <font color="Maroon" title="c8">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c9">K</font> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:89"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
( <font color="Maroon" title="c14">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c14">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c12">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c13">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">N</font> )
 <span class="kw">and </span><br/><a NAME="E2:89"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">b</font>
 <span class="kw">and </span><br/><a NAME="E3:89"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
( <font color="Maroon" title="c12">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K</font> &amp; <font color="Maroon" title="c13">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K</font> )
 <span class="kw">and </span><br/><a NAME="E4:89"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E5:89"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E6:89"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
<font color="Maroon" title="c9">K</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c9">K</font></span><br/>

<a NAME="E7:89"/>
<font color="Maroon" title="c8">A</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89"><span class="lab"><font color="Green" title="E62">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">q</font>, <font color="Maroon" title="c16">r</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="E9:89"/><span class="lab"><font color="Green" title="E65">A7</font></span>: 
<font color="Maroon" title="c15">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">r</font>
 <span class="kw">and </span><br/><a NAME="E10:89"/><span class="lab"><font color="Green" title="E66">A8</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analmetr.html#K3" title="ANALMETR:func.3">Line</a> (<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c16">r</font>)
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">q9</font> = <font color="Maroon" title="c15">q</font>, <font color="Maroon" title="c18">r9</font> = <font color="Maroon" title="c16">r</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">POS</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c7">POS</font> #) ;<br/>
<a NAME="E12:89"/>
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Line</a> (<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c16">r</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="c17">q9</font>,<font color="Maroon" title="c18">r9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T41" target="_self" onmouseover="rs('analmetr/T41')" onmouseout="rh()">Th41</a></span>;<br/>
<a NAME="E13:89"/><span class="kw">then </span>
( <font color="Maroon" title="c15">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c16">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:89"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="ref" href="aff_1.html#T15" onmouseover="rs('aff_1/T15')" onmouseout="rh()">AFF_1:15</a></span>;<br/>
<a NAME="E14:89"/><span class="kw">then </span>
( <font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c16">r</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c16">r</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c13">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E4:89"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="analmetr.html#T56" target="_self" onmouseover="rs('analmetr/T56')" onmouseout="rh()">Th56</a></span>;<br/>
<a NAME="E15:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A9</font></span>: 
<font color="Maroon" title="c15">q</font>,<font color="Maroon" title="c16">r</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</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="ref" href="analmetr.html#D8" target="_self" onmouseover="rs('analmetr/D8')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E16:89"/>
<font color="Maroon" title="c9">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analmetr.html#K3" title="ANALMETR:func.3">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="#E2:89"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="ref" href="analmetr.html#T54" target="_self" onmouseover="rs('analmetr/T54')" onmouseout="rh()">Th54</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:89"/>
<font color="Maroon" title="c8">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c9">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E9:89"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E10:89"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="txt" href="#E15:89"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="ref" href="analmetr.html#T45" target="_self" onmouseover="rs('analmetr/T45')" onmouseout="rh()">Th45</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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