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<b>let </b><font color="Maroon" title="c7">POS</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="analmetr.html#L1" title="ANALMETR:struct.1">ParOrtStr</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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font><br/> for <font color="Olive" title="b3">K</font>, <font color="Olive" title="b4">M</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">POS</font> holds <br/> ( ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Olive" title="b3">K</font> implies <font color="Olive" title="b3">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> ) &amp; ( <font color="Olive" title="b3">K</font> <a href="analmetr.html#R7" title="ANALMETR:pred.7">_|_</a> <font color="Olive" title="b4">M</font> implies ( <font color="Olive" title="b3">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> &amp; <font color="Olive" title="b4">M</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> ) ) )</span><br/><b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">M</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">POS</font> holds <br/> ( ( <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Olive" title="b1">K</font> implies <font color="Olive" title="b1">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> ) &amp; ( <font color="Olive" title="b1">K</font> <a href="analmetr.html#R7" title="ANALMETR:pred.7">_|_</a> <font color="Olive" title="b2">M</font> implies ( <font color="Olive" title="b1">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> &amp; <font color="Olive" title="b2">M</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> ) ) )</span><br/><b>let </b><font color="Maroon" title="c10">K</font>, <font color="Maroon" title="c11">M</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Maroon" title="c10">K</font> implies <font color="Maroon" title="c10">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> ) &amp; ( <font color="Maroon" title="c10">K</font> <a href="analmetr.html#R7" title="ANALMETR:pred.7">_|_</a> <font color="Maroon" title="c11">M</font> implies ( <font color="Maroon" title="c10">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> &amp; <font color="Maroon" title="c11">M</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> ) ) )</span><br/>









<div><i><font color="Green" title="E59">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">POS</font>  st <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Olive" title="b1">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a></span><br/><b>let </b><font color="Maroon" title="c14">K</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Maroon" title="c14">K</font> implies <font color="Maroon" title="c14">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> )</span><br/><b>assume </b><a NAME="E1:74_1"/>
<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Maroon" title="c14">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a></span><br/><a NAME="E2:74_1"/><b>then </b>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c14">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#D14" target="_self" title="ANALMETR:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E3:74_1"/>
<font color="Maroon" title="c14">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>
 <b>by </b><i><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/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_2"/>
<font color="Maroon" title="c10">K</font> <a href="analmetr.html#R7" title="ANALMETR:pred.7">_|_</a> <font color="Maroon" title="c11">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> &amp; <font color="Maroon" title="c11">M</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> )</span><br/><a NAME="E2:74_2"/><b>then </b><i><font color="Green" title="E60">A2</font></i>: 
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c10">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Maroon" title="c11">M</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#D15" target="_self" title="ANALMETR:def.15">Def15</a></i>;<br/><b>hence </b><a NAME="E3:74_2"/>
<font color="Maroon" title="c10">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>
 <b>by </b><i><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"> <font color="Maroon" title="c11">M</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a></span><br/><b>thus </b><a NAME="E4:74_2"/>
<font color="Maroon" title="c11">M</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>
 <b>by </b><i><a class="txt" href="#E1:74"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E2:74_2"><i><font color="Green" title="E60">A2</font></i></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>hence </b><a NAME="E3:74"/>
( ( <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R6" title="ANALMETR:pred.6">_|_</a> <font color="Maroon" title="c10">K</font> implies <font color="Maroon" title="c10">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> ) &amp; ( <font color="Maroon" title="c10">K</font> <a href="analmetr.html#R7" title="ANALMETR:pred.7">_|_</a> <font color="Maroon" title="c11">M</font> implies ( <font color="Maroon" title="c10">K</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> &amp; <font color="Maroon" title="c11">M</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> ) ) )
 <b>by </b><i><a class="txt" href="#E1:74"><i><font color="Green" title="E59">A1</font></i></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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