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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</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="c1">X</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>  iff <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V6" title="CONMETR1:attr.6">satisfying_major_indirect_Scherungssatz</a>  )</span><br/>

<a NAME="E1:16"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
( <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>  implies <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V6" title="CONMETR1:attr.6">satisfying_major_indirect_Scherungssatz</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</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="c1">X</font> is  <a href="conmetr1.html#V6" title="CONMETR1:attr.6">satisfying_major_indirect_Scherungssatz</a> </span><br/>

<a NAME="E2:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V4" title="AFF_2:attr.4">Desarguesian</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_2.html#T11" onmouseover="rs('aff_2/T11')" onmouseout="rh()">AFF_2:11</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a1</font>, <font color="Olive" title="b3">a2</font>, <font color="Olive" title="b4">a3</font>, <font color="Olive" title="b5">a4</font>, <font color="Olive" title="b6">b1</font>, <font color="Olive" title="b7">b2</font>, <font color="Olive" title="b8">b3</font>, <font color="Olive" title="b9">b4</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b10">M</font>, <font color="Olive" title="b11">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b10">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b11">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; <font color="Olive" title="b2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; <font color="Olive" title="b5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; <font color="Olive" title="b6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; <font color="Olive" title="b8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp;  not <font color="Olive" title="b5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp;  not <font color="Olive" title="b3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp;  not <font color="Olive" title="b6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp;  not <font color="Olive" title="b8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp;  not <font color="Olive" title="b2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp;  not <font color="Olive" title="b4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp;  not <font color="Olive" title="b7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp;  not <font color="Olive" title="b9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; <font color="Olive" title="b4">a3</font>,<font color="Olive" title="b3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">b3</font>,<font color="Olive" title="b7">b2</font> &amp; <font color="Olive" title="b3">a2</font>,<font color="Olive" title="b2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b2</font>,<font color="Olive" title="b6">b1</font> &amp; <font color="Olive" title="b2">a1</font>,<font color="Olive" title="b5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b1</font>,<font color="Olive" title="b9">b4</font> holds <br/><font color="Olive" title="b4">a3</font>,<font color="Olive" title="b5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">b3</font>,<font color="Olive" title="b9">b4</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">a2</font>, <font color="Maroon" title="c5">a3</font>, <font color="Maroon" title="c6">a4</font>, <font color="Maroon" title="c7">b1</font>, <font color="Maroon" title="c8">b2</font>, <font color="Maroon" title="c9">b3</font>, <font color="Maroon" title="c10">b4</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</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">M</font>, <font color="Olive" title="b2">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c5">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c10">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c6">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c9">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp;  not <font color="Maroon" title="c6">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  not <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  not <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  not <font color="Maroon" title="c9">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  not <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp;  not <font color="Maroon" title="c5">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp;  not <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp;  not <font color="Maroon" title="c10">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c4">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c8">b2</font> &amp; <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c10">b4</font> holds <br/><font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">M</font>, <font color="Maroon" title="c12">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</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="c11">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c12">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c5">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c10">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Maroon" title="c6">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Maroon" title="c9">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp;  not <font color="Maroon" title="c6">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp;  not <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp;  not <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp;  not <font color="Maroon" title="c9">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp;  not <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp;  not <font color="Maroon" title="c5">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp;  not <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp;  not <font color="Maroon" title="c10">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c4">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c8">b2</font> &amp; <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c10">b4</font> implies <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16_1_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c11">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:16_1_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c12">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:16_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">and </span><br/><a NAME="E5:16_1_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E6:16_1_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c5">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E7:16_1_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E8:16_1_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c10">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E9:16_1_1"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c4">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">and </span><br/><a NAME="E10:16_1_1"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c6">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">and </span><br/><a NAME="E11:16_1_1"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">and </span><br/><a NAME="E12:16_1_1"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c9">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">and </span><br/><a NAME="E13:16_1_1"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
 not <font color="Maroon" title="c6">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E14:16_1_1"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
 not <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E15:16_1_1"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
 not <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E16:16_1_1"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
 not <font color="Maroon" title="c9">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E17:16_1_1"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
 not <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">and </span><br/><a NAME="E18:16_1_1"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
 not <font color="Maroon" title="c5">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">and </span><br/><a NAME="E19:16_1_1"/><span class="lab"><font color="Green" title="E30">A22</font></span>: 
 not <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">and </span><br/><a NAME="E20:16_1_1"/><span class="lab"><font color="Green" title="E31">A23</font></span>: 
 not <font color="Maroon" title="c10">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">and </span><br/><a NAME="E21:16_1_1"/><span class="lab"><font color="Green" title="E32">A24</font></span>: 
<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c4">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c8">b2</font>
 <span class="kw">and </span><br/><a NAME="E22:16_1_1"/><span class="lab"><font color="Green" title="E33">A25</font></span>: 
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c7">b1</font>
 <span class="kw">and </span><br/><a NAME="E23:16_1_1"/><span class="lab"><font color="Green" title="E34">A26</font></span>: 
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c10">b4</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="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font></span><br/><div><span class="lab"><font color="Green" title="E35">A27</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><span class="kw">now </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"> ( <font color="Maroon" title="c3">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a3</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a4</font> implies <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font> )</span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16_1_1_1"/><span class="lab"><font color="Green" title="E36">A28</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a3</font>
 <span class="kw">and </span><br/><a NAME="E2:16_1_1_1"/><span class="lab"><font color="Green" title="E37">A29</font></span>: 
<font color="Maroon" title="c4">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a4</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="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font></span><br/><a NAME="E3:16_1_1_1"/>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_1_1_1_1"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="aff_1.html#T39" onmouseover="rs('aff_1/T39')" onmouseout="rh()">AFF_1:39</a>, <a class="ref" href="aff_1.html#T41" onmouseover="rs('aff_1/T41')" onmouseout="rh()">AFF_1:41</a></span>;<br/>
<a NAME="E2:16_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A30</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:16_1_1_1_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  holds <br/>(  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> or  not <font color="Olive" title="b2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</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"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E4:16_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E2:16_1_1_1_1"><span class="lab"><font color="Green" title="E38">A30</font></span></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/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:16_1_1_1"/><span class="lab"><font color="Green" title="E38">A31</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c13">d</font>
 <span class="kw">and </span><br/><a NAME="E6:16_1_1_1"/><span class="lab"><font color="Green" title="E39">A32</font></span>: 
<font color="Maroon" title="c4">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">d</font>
 <span class="kw">and </span><br/><a NAME="E7:16_1_1_1"/><span class="lab"><font color="Green" title="E40">A33</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="translac.html#T1" onmouseover="rs('translac/T1')" onmouseout="rh()">TRANSLAC:1</a></span>;<br/><a NAME="E8:16_1_1_1"/><span class="lab"><font color="Green" title="E41">A34</font></span>: 
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_1"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E9:16_1_1_1"/><span class="lab"><font color="Green" title="E42">A35</font></span>: 
 ex <font color="Olive" title="b1">c2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Olive" title="b1">c2</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c14">e2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E2:16_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A36</font></span>: 
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c14">e2</font>
 <span class="kw">and </span><br/><a NAME="E3:16_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A37</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" onmouseover="rs('diraf/T40')" onmouseout="rh()">DIRAF:40</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">e1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:16_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A38</font></span>: 
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c15">e1</font>
 <span class="kw">and </span><br/><a NAME="E6:16_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A39</font></span>: 
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" onmouseover="rs('diraf/T40')" onmouseout="rh()">DIRAF:40</a></span>;<br/>
<a NAME="E7:16_1_1_1_2"/>
 not <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c15">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c14">e2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_2_1"/>
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c15">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c14">e2</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"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c14">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_1_2"><span class="lab"><font color="Green" title="E45">A38</font></span></a>, <a class="txt" href="#E6:16_1_1_1_2"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E3:16_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1_2"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="txt" href="#E3:16_1_1_1_2"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E4:16_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c4">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/>
<a NAME="E5:16_1_1_1_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:16_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">c2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E9:16_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A40</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c15">e1</font>,<font color="Maroon" title="c16">c2</font>
 <span class="kw">and </span><br/><a NAME="E10:16_1_1_1_2"/><span class="lab"><font color="Green" title="E48">A41</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c14">e2</font>,<font color="Maroon" title="c16">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T60" onmouseover="rs('aff_1/T60')" onmouseout="rh()">AFF_1:60</a></span>;<br/>
<a NAME="E11:16_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c14">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c16">c2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1_1_2"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E12:16_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c16">c2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1_2"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="txt" href="#E3:16_1_1_1_2"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E13:16_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A42</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c16">c2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">c2</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="c16">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c16">c2</font> )</span><br/>

<a NAME="E14:16_1_1_1_2"/>
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c15">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c16">c2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_1_1_2"><span class="lab"><font color="Green" title="E47">A40</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:16_1_1_1_2"/>
( <font color="Maroon" title="c16">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c16">c2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E5:16_1_1_1_2"><span class="lab"><font color="Green" title="E45">A38</font></span></a>, <a class="txt" href="#E6:16_1_1_1_2"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="txt" href="#E13:16_1_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/>


</div><span class="kw">end;</span></div><a NAME="E10:16_1_1_1"/><span class="lab"><font color="Green" title="E43">A43</font></span>: 
 ex <font color="Olive" title="b1">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Olive" title="b1">c1</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c14">e2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E2:16_1_1_1_3"/><span class="lab"><font color="Green" title="E44">A44</font></span>: 
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c14">e2</font>
 <span class="kw">and </span><br/><a NAME="E3:16_1_1_1_3"/><span class="lab"><font color="Green" title="E45">A45</font></span>: 
<font color="Maroon" title="c4">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" onmouseover="rs('diraf/T40')" onmouseout="rh()">DIRAF:40</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">e1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:16_1_1_1_3"/><span class="lab"><font color="Green" title="E46">A46</font></span>: 
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c15">e1</font>
 <span class="kw">and </span><br/><a NAME="E6:16_1_1_1_3"/><span class="lab"><font color="Green" title="E47">A47</font></span>: 
<font color="Maroon" title="c8">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" onmouseover="rs('diraf/T40')" onmouseout="rh()">DIRAF:40</a></span>;<br/>
<a NAME="E7:16_1_1_1_3"/>
 not <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c15">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c14">e2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_3_1"/>
<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c15">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c14">e2</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"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c14">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_1_3"><span class="lab"><font color="Green" title="E46">A46</font></span></a>, <a class="txt" href="#E6:16_1_1_1_3"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E3:16_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">a4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1_3"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="txt" href="#E3:16_1_1_1_3"><span class="lab"><font color="Green" title="E45">A45</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E4:16_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a4</font>,<font color="Maroon" title="c4">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a4</font>,<font color="Maroon" title="c3">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/>
<a NAME="E5:16_1_1_1_3_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a4</font>,<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:16_1_1_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E2:16_1_1_1"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E9:16_1_1_1_3"/><span class="lab"><font color="Green" title="E48">A48</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c15">e1</font>,<font color="Maroon" title="c16">c1</font>
 <span class="kw">and </span><br/><a NAME="E10:16_1_1_1_3"/><span class="lab"><font color="Green" title="E49">A49</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c14">e2</font>,<font color="Maroon" title="c16">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T60" onmouseover="rs('aff_1/T60')" onmouseout="rh()">AFF_1:60</a></span>;<br/>
<a NAME="E11:16_1_1_1_3"/>
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c14">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c16">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1_1_3"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E12:16_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c16">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1_3"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="txt" href="#E3:16_1_1_1_3"><span class="lab"><font color="Green" title="E45">A45</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E13:16_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A50</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">a4</font>,<font color="Maroon" title="c16">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">c1</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="c16">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c16">c1</font> )</span><br/>

<a NAME="E14:16_1_1_1_3"/>
<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c15">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c16">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_1_1_3"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:16_1_1_1_3"/>
( <font color="Maroon" title="c16">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c16">c1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E2:16_1_1_1"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E5:16_1_1_1_3"><span class="lab"><font color="Green" title="E46">A46</font></span></a>, <a class="txt" href="#E6:16_1_1_1_3"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="txt" href="#E13:16_1_1_1_3"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c14">c2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E12:16_1_1_1"/><span class="lab"><font color="Green" title="E44">A51</font></span>: 
<font color="Maroon" title="c14">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E13:16_1_1_1"/><span class="lab"><font color="Green" title="E45">A52</font></span>: 
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c14">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_1_1"><span class="lab"><font color="Green" title="E42">A35</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E15:16_1_1_1"/><span class="lab"><font color="Green" title="E46">A53</font></span>: 
<font color="Maroon" title="c15">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">and </span><br/><a NAME="E16:16_1_1_1"/><span class="lab"><font color="Green" title="E47">A54</font></span>: 
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c15">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1_1"><span class="lab"><font color="Green" title="E43">A43</font></span></a></span>;<br/><a NAME="E17:16_1_1_1"/>
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c10">b4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c15">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E23:16_1_1"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E16:16_1_1_1"><span class="lab"><font color="Green" title="E47">A54</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E18:16_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A55</font></span>: 
<font color="Maroon" title="c10">b4</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c15">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E19:16_1_1_1"/><span class="lab"><font color="Green" title="E49">A56</font></span>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c1</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">c2</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_4"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E50">A57</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_4_1"><span class="kw">now </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">  not <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c2</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1_1_1_4_1"/>
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c2</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"> contradiction</span><br/><a NAME="E2:16_1_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A58</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16_1_1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E3:16_1_1_1_4_1"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">a4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="aff_1.html#T39" onmouseover="rs('aff_1/T39')" onmouseout="rh()">AFF_1:39</a>, <a class="ref" href="aff_1.html#T41" onmouseover="rs('aff_1/T41')" onmouseout="rh()">AFF_1:41</a></span>;<br/><a NAME="E4:16_1_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E2:16_1_1_1_4_1"><span class="lab"><font color="Green" title="E51">A58</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E5:16_1_1_1_4_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">a4</font>,<font color="Maroon" title="c5">a3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:16_1_1_1_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E2:16_1_1_1"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E51">A59</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_4_2"><span class="kw">now </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">  not <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c1</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1_1_1_4_2"/>
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c1</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"> contradiction</span><br/><a NAME="E2:16_1_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A60</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16_1_1_1"><span class="lab"><font color="Green" title="E47">A54</font></span></a>, <a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E3:16_1_1_1_4_2"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="aff_1.html#T39" onmouseover="rs('aff_1/T39')" onmouseout="rh()">AFF_1:39</a>, <a class="ref" href="aff_1.html#T41" onmouseover="rs('aff_1/T41')" onmouseout="rh()">AFF_1:41</a></span>;<br/><a NAME="E4:16_1_1_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E19:16_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E2:16_1_1_1_4_2"><span class="lab"><font color="Green" title="E52">A60</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E5:16_1_1_1_4_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:16_1_1_1_4_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:16_1_1_1_4"/>
(  not <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c1</font> or  not <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">c2</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"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E4:16_1_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1_4"><span class="lab"><font color="Green" title="E51">A59</font></span></a>, <a class="txt" href="#E1:16_1_1_1_4"><span class="lab"><font color="Green" title="E50">A57</font></span></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/>


</div><span class="kw">end;</span></div><a NAME="E20:16_1_1_1"/>
<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c4">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">c2</font>,<font color="Maroon" title="c7">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16_1_1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E21:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">c2</font>,<font color="Maroon" title="c7">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E21:16_1_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E22:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c9">b3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">c2</font>,<font color="Maroon" title="c7">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E23:16_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A61</font></span>: 
<font color="Maroon" title="c10">b4</font>,<font color="Maroon" title="c9">b3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">c2</font>,<font color="Maroon" title="c15">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E16:16_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E19:16_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E20:16_1_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E46">A53</font></span></a>, <a class="txt" href="#E12:16_1_1_1"><span class="lab"><font color="Green" title="E44">A51</font></span></a>, <a class="txt" href="#E19:16_1_1_1"><span class="lab"><font color="Green" title="E49">A56</font></span></a>, <a class="txt" href="#E18:16_1_1_1"><span class="lab"><font color="Green" title="E48">A55</font></span></a>, <a class="ref" href="aff_2.html#D2" onmouseover="rs('aff_2/D2')" onmouseout="rh()">AFF_2:def 2</a></span>;<br/><a NAME="E24:16_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_1"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/><a NAME="E25:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E26:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E27:16_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A62</font></span>: 
<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E22:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E28:16_1_1_1"/><span class="lab"><font color="Green" title="E52">A63</font></span>: 
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b3</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_5"/>
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b3</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"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16_1_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/>
<a NAME="E3:16_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A64</font></span>: 
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E22:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E4:16_1_1_1_5"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="aff_1.html#T39" onmouseover="rs('aff_1/T39')" onmouseout="rh()">AFF_1:39</a>, <a class="ref" href="aff_1.html#T41" onmouseover="rs('aff_1/T41')" onmouseout="rh()">AFF_1:41</a></span>;<br/>
<a NAME="E5:16_1_1_1_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:16_1_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E3:16_1_1_1_5"><span class="lab"><font color="Green" title="E53">A64</font></span></a>, <a class="ref" href="aff_1.html#T14" onmouseover="rs('aff_1/T14')" onmouseout="rh()">AFF_1:14</a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/>


</div><span class="kw">end;</span></div><a NAME="E29:16_1_1_1"/><span class="lab"><font color="Green" title="E53">A65</font></span>: 
<font color="Maroon" title="c15">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">c2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_6"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_1_1_1_6"/><span class="lab"><font color="Green" title="E54">A66</font></span>: 
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_6_1"/>
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c1</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"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E22:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E16:16_1_1_1"><span class="lab"><font color="Green" title="E47">A54</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E3:16_1_1_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/>
<a NAME="E4:16_1_1_1_6_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font>,<font color="Maroon" title="c4">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E5:16_1_1_1_6_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">a4</font>,<font color="Maroon" title="c3">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:16_1_1_1_6_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E2:16_1_1_1"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:16_1_1_1_6"/>
<font color="Maroon" title="c15">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c2</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"> contradiction</span><br/>

<a NAME="E3:16_1_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c4">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c15">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16_1_1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/>
<a NAME="E4:16_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A67</font></span>: 
<font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c15">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E21:16_1_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E5:16_1_1_1_6"/>
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c15">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c7">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E46">A53</font></span></a>, <a class="ref" href="aff_1.html#T39" onmouseover="rs('aff_1/T39')" onmouseout="rh()">AFF_1:39</a>, <a class="ref" href="aff_1.html#T41" onmouseover="rs('aff_1/T41')" onmouseout="rh()">AFF_1:41</a></span>;<br/>
<a NAME="E6:16_1_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c8">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_1_1_6"><span class="lab"><font color="Green" title="E55">A67</font></span></a>, <a class="txt" href="#E1:16_1_1_1_6"><span class="lab"><font color="Green" title="E54">A66</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E7:16_1_1_1_6"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:16_1_1_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E19:16_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E28:16_1_1_1"><span class="lab"><font color="Green" title="E52">A63</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/>


</div><span class="kw">end;</span></div><a NAME="E30:16_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">d</font>,<font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T7" onmouseover="rs('aff_1/T7')" onmouseout="rh()">AFF_1:7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E32:16_1_1_1"/><span class="lab"><font color="Green" title="E54">A68</font></span>: 
<font color="Maroon" title="c16">Y</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E33:16_1_1_1"/><span class="lab"><font color="Green" title="E55">A69</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y</font>
 <span class="kw">and </span><br/><a NAME="E34:16_1_1_1"/><span class="lab"><font color="Green" title="E56">A70</font></span>: 
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y</font>
 <span class="kw">and </span><br/><a NAME="E35:16_1_1_1"/>
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T21" onmouseover="rs('aff_1/T21')" onmouseout="rh()">AFF_1:21</a></span>;<br/><a NAME="E36:16_1_1_1"/><span class="lab"><font color="Green" title="E57">A71</font></span>: 
( <font color="Maroon" title="c11">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Maroon" title="c11">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Y</font> &amp; <font color="Maroon" title="c12">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Y</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_7"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E58">A72</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_7_1"><span class="kw">now </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">  not <font color="Maroon" title="c12">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Y</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1_1_1_7_1"/><span class="lab"><font color="Green" title="E59">A73</font></span>: 
<font color="Maroon" title="c12">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Y</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"> contradiction</span><br/><a NAME="E2:16_1_1_1_7_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c13">d</font>,<font color="Maroon" title="c3">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_1"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_1_1_7_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="txt" href="#E34:16_1_1_1"><span class="lab"><font color="Green" title="E56">A70</font></span></a>, <a class="txt" href="#E1:16_1_1_1_7_1"><span class="lab"><font color="Green" title="E59">A73</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E59">A74</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_7_2"><span class="kw">now </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">  not <font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Y</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1_1_1_7_2"/><span class="lab"><font color="Green" title="E60">A75</font></span>: 
<font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Y</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"> contradiction</span><br/><a NAME="E2:16_1_1_1_7_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c13">d</font>,<font color="Maroon" title="c4">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_1"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_1_1_7_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E7:16_1_1_1"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="txt" href="#E34:16_1_1_1"><span class="lab"><font color="Green" title="E56">A70</font></span></a>, <a class="txt" href="#E1:16_1_1_1_7_2"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:16_1_1_1_7"/>
(  not <font color="Maroon" title="c11">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">N</font> or  not <font color="Maroon" title="c11">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Y</font> or  not <font color="Maroon" title="c12">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Y</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"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E4:16_1_1_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E2:16_1_1_1_7"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E1:16_1_1_1_7"><span class="lab"><font color="Green" title="E58">A72</font></span></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/>


</div><span class="kw">end;</span></div><a NAME="E37:16_1_1_1"/><span class="lab"><font color="Green" title="E58">A76</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">d</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_8"/>
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">d</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"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_8"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_1"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:16_1_1_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/>


</div><span class="kw">end;</span></div><a NAME="E38:16_1_1_1"/>
 ex <font color="Olive" title="b1">d1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font>,<font color="Olive" title="b1">d1</font> &amp; <font color="Olive" title="b1">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c17">e2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E2:16_1_1_1_9"/><span class="lab"><font color="Green" title="E59">A77</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">e2</font>
 <span class="kw">and </span><br/><a NAME="E3:16_1_1_1_9"/><span class="lab"><font color="Green" title="E60">A78</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" onmouseover="rs('diraf/T40')" onmouseout="rh()">DIRAF:40</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">e1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:16_1_1_1_9"/><span class="lab"><font color="Green" title="E61">A79</font></span>: 
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c18">e1</font>
 <span class="kw">and </span><br/><a NAME="E6:16_1_1_1_9"/><span class="lab"><font color="Green" title="E62">A80</font></span>: 
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" onmouseover="rs('diraf/T40')" onmouseout="rh()">DIRAF:40</a></span>;<br/>
<a NAME="E7:16_1_1_1_9"/>
 not <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c18">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">e2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_1_1_1_9_1"/>
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c18">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">e2</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"> contradiction</span><br/>

<a NAME="E2:16_1_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_1_9"><span class="lab"><font color="Green" title="E61">A79</font></span></a>, <a class="txt" href="#E6:16_1_1_1_9"><span class="lab"><font color="Green" title="E62">A80</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E3:16_1_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1_9"><span class="lab"><font color="Green" title="E59">A77</font></span></a>, <a class="txt" href="#E3:16_1_1_1_9"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E4:16_1_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/>
<a NAME="E5:16_1_1_1_9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A81</font></span>: 
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E22:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E6:16_1_1_1_9_1"/>
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c13">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_1"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E7:16_1_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c13">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E5:16_1_1_1_9_1"><span class="lab"><font color="Green" title="E63">A81</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E8:16_1_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">d</font>,<font color="Maroon" title="c4">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/>
<a NAME="E9:16_1_1_1_9_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">d</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E10:16_1_1_1_9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A82</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c13">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/>
<a NAME="E11:16_1_1_1_9_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c13">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_1"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/>
<a NAME="E12:16_1_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c13">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:16_1_1_1_9_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="txt" href="#E10:16_1_1_1_9_1"><span class="lab"><font color="Green" title="E64">A82</font></span></a>, <a class="ref" href="aff_1.html#T14" onmouseover="rs('aff_1/T14')" onmouseout="rh()">AFF_1:14</a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">d1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E9:16_1_1_1_9"/><span class="lab"><font color="Green" title="E63">A83</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c18">e1</font>,<font color="Maroon" title="c19">d1</font>
 <span class="kw">and </span><br/><a NAME="E10:16_1_1_1_9"/><span class="lab"><font color="Green" title="E64">A84</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">e2</font>,<font color="Maroon" title="c19">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T60" onmouseover="rs('aff_1/T60')" onmouseout="rh()">AFF_1:60</a></span>;<br/>
<a NAME="E11:16_1_1_1_9"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c17">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c19">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1_1_9"><span class="lab"><font color="Green" title="E64">A84</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E12:16_1_1_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c19">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1_9"><span class="lab"><font color="Green" title="E59">A77</font></span></a>, <a class="txt" href="#E3:16_1_1_1_9"><span class="lab"><font color="Green" title="E60">A78</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<a NAME="E13:16_1_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A85</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">d</font>,<font color="Maroon" title="c19">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c19">d1</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"> (  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c19">d1</font> &amp; <font color="Maroon" title="c19">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y</font> )</span><br/>

<a NAME="E14:16_1_1_1_9"/>
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c18">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c19">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_1_1_9"><span class="lab"><font color="Green" title="E63">A83</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E15:16_1_1_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c19">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_1_9"><span class="lab"><font color="Green" title="E61">A79</font></span></a>, <a class="txt" href="#E6:16_1_1_1_9"><span class="lab"><font color="Green" title="E62">A80</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:16_1_1_1_9"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c19">d1</font> &amp; <font color="Maroon" title="c19">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:16_1_1_1"><span class="lab"><font color="Green" title="E58">A76</font></span></a>, <a class="txt" href="#E32:16_1_1_1"><span class="lab"><font color="Green" title="E54">A68</font></span></a>, <a class="txt" href="#E33:16_1_1_1"><span class="lab"><font color="Green" title="E55">A69</font></span></a>, <a class="txt" href="#E34:16_1_1_1"><span class="lab"><font color="Green" title="E56">A70</font></span></a>, <a class="txt" href="#E13:16_1_1_1_9"><span class="lab"><font color="Green" title="E65">A85</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</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/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">d1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E40:16_1_1_1"/><span class="lab"><font color="Green" title="E59">A86</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c17">d1</font>
 <span class="kw">and </span><br/><a NAME="E41:16_1_1_1"/><span class="lab"><font color="Green" title="E60">A87</font></span>: 
<font color="Maroon" title="c17">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y</font>
 ;<br/><a NAME="E42:16_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c17">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:16_1_1_1"><span class="lab"><font color="Green" title="E59">A86</font></span></a>, <a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/><a NAME="E43:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c17">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E44:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c13">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c17">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E27:16_1_1_1"><span class="lab"><font color="Green" title="E51">A62</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E45:16_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A88</font></span>: 
<font color="Maroon" title="c13">d</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">d1</font>,<font color="Maroon" title="c15">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E37:16_1_1_1"><span class="lab"><font color="Green" title="E58">A76</font></span></a>, <a class="txt" href="#E32:16_1_1_1"><span class="lab"><font color="Green" title="E54">A68</font></span></a>, <a class="txt" href="#E33:16_1_1_1"><span class="lab"><font color="Green" title="E55">A69</font></span></a>, <a class="txt" href="#E34:16_1_1_1"><span class="lab"><font color="Green" title="E56">A70</font></span></a>, <a class="txt" href="#E41:16_1_1_1"><span class="lab"><font color="Green" title="E60">A87</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E46">A53</font></span></a>, <a class="txt" href="#E16:16_1_1_1"><span class="lab"><font color="Green" title="E47">A54</font></span></a>, <a class="txt" href="#E36:16_1_1_1"><span class="lab"><font color="Green" title="E57">A71</font></span></a>, <a class="ref" href="aff_2.html#D4" onmouseover="rs('aff_2/D4')" onmouseout="rh()">AFF_2:def 4</a></span>;<br/><a NAME="E46:16_1_1_1"/>
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c17">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:16_1_1_1"><span class="lab"><font color="Green" title="E59">A86</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><a NAME="E47:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c17">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E48:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c17">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E22:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E49:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c13">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c17">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E8:16_1_1_1"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><a NAME="E50:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font>,<font color="Maroon" title="c5">a3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">d1</font>,<font color="Maroon" title="c14">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E9:16_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E37:16_1_1_1"><span class="lab"><font color="Green" title="E58">A76</font></span></a>, <a class="txt" href="#E32:16_1_1_1"><span class="lab"><font color="Green" title="E54">A68</font></span></a>, <a class="txt" href="#E33:16_1_1_1"><span class="lab"><font color="Green" title="E55">A69</font></span></a>, <a class="txt" href="#E34:16_1_1_1"><span class="lab"><font color="Green" title="E56">A70</font></span></a>, <a class="txt" href="#E41:16_1_1_1"><span class="lab"><font color="Green" title="E60">A87</font></span></a>, <a class="txt" href="#E12:16_1_1_1"><span class="lab"><font color="Green" title="E44">A51</font></span></a>, <a class="txt" href="#E13:16_1_1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E36:16_1_1_1"><span class="lab"><font color="Green" title="E57">A71</font></span></a>, <a class="ref" href="aff_2.html#D4" onmouseover="rs('aff_2/D4')" onmouseout="rh()">AFF_2:def 4</a></span>;<br/><a NAME="E51:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">c2</font>,<font color="Maroon" title="c15">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E37:16_1_1_1"><span class="lab"><font color="Green" title="E58">A76</font></span></a>, <a class="txt" href="#E32:16_1_1_1"><span class="lab"><font color="Green" title="E54">A68</font></span></a>, <a class="txt" href="#E33:16_1_1_1"><span class="lab"><font color="Green" title="E55">A69</font></span></a>, <a class="txt" href="#E34:16_1_1_1"><span class="lab"><font color="Green" title="E56">A70</font></span></a>, <a class="txt" href="#E41:16_1_1_1"><span class="lab"><font color="Green" title="E60">A87</font></span></a>, <a class="txt" href="#E15:16_1_1_1"><span class="lab"><font color="Green" title="E46">A53</font></span></a>, <a class="txt" href="#E12:16_1_1_1"><span class="lab"><font color="Green" title="E44">A51</font></span></a>, <a class="txt" href="#E36:16_1_1_1"><span class="lab"><font color="Green" title="E57">A71</font></span></a>, <a class="txt" href="#E45:16_1_1_1"><span class="lab"><font color="Green" title="E61">A88</font></span></a>, <a class="ref" href="aff_2.html#D4" onmouseover="rs('aff_2/D4')" onmouseout="rh()">AFF_2:def 4</a></span>;<br/><a NAME="E52:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">b4</font>,<font color="Maroon" title="c9">b3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:16_1_1_1"><span class="lab"><font color="Green" title="E53">A65</font></span></a>, <a class="txt" href="#E23:16_1_1_1"><span class="lab"><font color="Green" title="E50">A61</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E53:16_1_1_1"/>
<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</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/></div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_2"><span class="kw">now </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"> ( ( <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a3</font> or <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a4</font> ) implies <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font> )</span></a><div class="add"><div><span class="lab"><font color="Green" title="E36">A89</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_2_1"><span class="kw">now </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"> ( <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a4</font> implies <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font> )</span></a><div class="add"><a NAME="E1:16_1_1_2_1"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="aff_1.html#T39" onmouseover="rs('aff_1/T39')" onmouseout="rh()">AFF_1:39</a>, <a class="ref" href="aff_1.html#T41" onmouseover="rs('aff_1/T41')" onmouseout="rh()">AFF_1:41</a></span>;<br/><a NAME="E2:16_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A90</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c10">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E3:16_1_1_2_1"/><span class="lab"><font color="Green" title="E38">A91</font></span>: 
<font color="Maroon" title="c4">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a4</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="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font></span><br/><a NAME="E4:16_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E5:16_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c10">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E13:16_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E23:16_1_1"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:16_1_1_2_1"/>
<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E19:16_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E21:16_1_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E3:16_1_1_2_1"><span class="lab"><font color="Green" title="E38">A91</font></span></a>, <a class="txt" href="#E2:16_1_1_2_1"><span class="lab"><font color="Green" title="E37">A90</font></span></a>, <a class="ref" href="aff_1.html#T14" onmouseover="rs('aff_1/T14')" onmouseout="rh()">AFF_1:14</a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E37">A92</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1_2_2"><span class="kw">now </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"> ( <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a3</font> implies <font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font> )</span></a><div class="add"><a NAME="E1:16_1_1_2_2"/>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">b3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E11:16_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="aff_1.html#T39" onmouseover="rs('aff_1/T39')" onmouseout="rh()">AFF_1:39</a>, <a class="ref" href="aff_1.html#T41" onmouseover="rs('aff_1/T41')" onmouseout="rh()">AFF_1:41</a></span>;<br/><a NAME="E2:16_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A93</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c9">b3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E3:16_1_1_2_2"/><span class="lab"><font color="Green" title="E39">A94</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a3</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="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font></span><br/><a NAME="E4:16_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c9">b3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16_1_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E5:16_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c9">b3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E22:16_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="aff_1.html#T5" onmouseover="rs('aff_1/T5')" onmouseout="rh()">AFF_1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:16_1_1_2_2"/>
<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E15:16_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E19:16_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E23:16_1_1"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E3:16_1_1_2_2"><span class="lab"><font color="Green" title="E39">A94</font></span></a>, <a class="txt" href="#E2:16_1_1_2_2"><span class="lab"><font color="Green" title="E38">A93</font></span></a>, <a class="ref" href="aff_1.html#T14" onmouseover="rs('aff_1/T14')" onmouseout="rh()">AFF_1:14</a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</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/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E3:16_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a3</font> or <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a4</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="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font></span><br/><span class="kw">hence </span><a NAME="E4:16_1_1_2"/>
<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_2"><span class="lab"><font color="Green" title="E37">A92</font></span></a>, <a class="txt" href="#E1:16_1_1_2"><span class="lab"><font color="Green" title="E36">A89</font></span></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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E26:16_1_1"/>
<font color="Maroon" title="c5">a3</font>,<font color="Maroon" title="c6">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:16_1_1"><span class="lab"><font color="Green" title="E35">A27</font></span></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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:16_1"/>
<font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V6" title="CONMETR1:attr.6">satisfying_major_indirect_Scherungssatz</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:16"/>
( <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V6" title="CONMETR1:attr.6">satisfying_major_indirect_Scherungssatz</a>  implies <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_2"/><span class="lab"><font color="Green" title="E10">A95</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V6" title="CONMETR1:attr.6">satisfying_major_indirect_Scherungssatz</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="c1">X</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_2_1"><span class="kw">now </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">  for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">o</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">a1</font>, <font color="Olive" title="b8">b1</font>, <font color="Olive" title="b9">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">a1</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">b</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">b1</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">c</font> &amp; <font color="Olive" title="b3">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">c1</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b8">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b9">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b8">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b7">a1</font> &amp; <font color="Olive" title="b5">b</font>,<font color="Olive" title="b9">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">c</font>,<font color="Olive" title="b8">b1</font> holds <br/><font color="Olive" title="b4">a</font>,<font color="Olive" title="b9">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">c</font>,<font color="Olive" title="b7">a1</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</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">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a1</font>, <font color="Olive" title="b6">b1</font>, <font color="Olive" title="b7">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">a1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">b1</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">c1</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b5">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b7">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">a1</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b7">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b6">b1</font> holds <br/><font color="Olive" title="b2">a</font>,<font color="Olive" title="b7">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">a1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">o</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c9">b1</font>, <font color="Maroon" title="c10">c1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</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="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a1</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b1</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c10">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a1</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">b1</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a1</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16_2_1"/><span class="lab"><font color="Green" title="E11">A96</font></span>: 
<font color="Maroon" title="c2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:16_2_1"/><span class="lab"><font color="Green" title="E12">A97</font></span>: 
<font color="Maroon" title="c3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:16_2_1"/><span class="lab"><font color="Green" title="E13">A98</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E4:16_2_1"/><span class="lab"><font color="Green" title="E14">A99</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E5:16_2_1"/><span class="lab"><font color="Green" title="E15">A100</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E6:16_2_1"/><span class="lab"><font color="Green" title="E16">A101</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">and </span><br/><a NAME="E7:16_2_1"/><span class="lab"><font color="Green" title="E17">A102</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a1</font>
 <span class="kw">and </span><br/><a NAME="E8:16_2_1"/><span class="lab"><font color="Green" title="E18">A103</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">and </span><br/><a NAME="E9:16_2_1"/><span class="lab"><font color="Green" title="E19">A104</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b1</font>
 <span class="kw">and </span><br/><a NAME="E10:16_2_1"/><span class="lab"><font color="Green" title="E20">A105</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font>
 <span class="kw">and </span><br/><a NAME="E11:16_2_1"/><span class="lab"><font color="Green" title="E21">A106</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c1</font>
 <span class="kw">and </span><br/><a NAME="E12:16_2_1"/><span class="lab"><font color="Green" title="E22">A107</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E13:16_2_1"/><span class="lab"><font color="Green" title="E23">A108</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E14:16_2_1"/><span class="lab"><font color="Green" title="E24">A109</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E15:16_2_1"/><span class="lab"><font color="Green" title="E25">A110</font></span>: 
<font color="Maroon" title="c8">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E16:16_2_1"/><span class="lab"><font color="Green" title="E26">A111</font></span>: 
<font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E17:16_2_1"/><span class="lab"><font color="Green" title="E27">A112</font></span>: 
<font color="Maroon" title="c10">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E18:16_2_1"/><span class="lab"><font color="Green" title="E28">A113</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">a1</font>
 <span class="kw">and </span><br/><a NAME="E19:16_2_1"/><span class="lab"><font color="Green" title="E29">A114</font></span>: 
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">b1</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="c5">a</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a1</font></span><br/><a NAME="E20:16_2_1"/><span class="lab"><font color="Green" title="E30">A115</font></span>: 
(  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp;  not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp;  not <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp;  not <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp;  not <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp;  not <font color="Maroon" title="c10">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_2_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_2_1_1"/><span class="lab"><font color="Green" title="E31">A116</font></span>: 
<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c10">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T2" onmouseover="rs('aff_1/T2')" onmouseout="rh()">AFF_1:2</a></span>;<br/>
<a NAME="E2:16_2_1_1"/><span class="lab"><font color="Green" title="E32">A117</font></span>: 
<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c9">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c9">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T2" onmouseover="rs('aff_1/T2')" onmouseout="rh()">AFF_1:2</a></span>;<br/>
<a NAME="E3:16_2_1_1"/><span class="lab"><font color="Green" title="E33">A118</font></span>: 
<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c8">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c8">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T2" onmouseover="rs('aff_1/T2')" onmouseout="rh()">AFF_1:2</a></span>;<br/>
<a NAME="E4:16_2_1_1"/><span class="lab"><font color="Green" title="E34">A119</font></span>: 
<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T2" onmouseover="rs('aff_1/T2')" onmouseout="rh()">AFF_1:2</a></span>;<br/>
<a NAME="E5:16_2_1_1"/><span class="lab"><font color="Green" title="E35">A120</font></span>: 
<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T2" onmouseover="rs('aff_1/T2')" onmouseout="rh()">AFF_1:2</a></span>;<br/>
<a NAME="E6:16_2_1_1"/><span class="lab"><font color="Green" title="E36">A121</font></span>: 
<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T2" onmouseover="rs('aff_1/T2')" onmouseout="rh()">AFF_1:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:16_2_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> or <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> or <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> or <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> or <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> or <font color="Maroon" title="c10">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</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"> contradiction</span><br/>

<a NAME="E8:16_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">M</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2_1"><span class="lab"><font color="Green" title="E11">A96</font></span></a>, <a class="txt" href="#E2:16_2_1"><span class="lab"><font color="Green" title="E12">A97</font></span></a>, <a class="txt" href="#E4:16_2_1"><span class="lab"><font color="Green" title="E14">A99</font></span></a>, <a class="txt" href="#E5:16_2_1"><span class="lab"><font color="Green" title="E15">A100</font></span></a>, <a class="txt" href="#E6:16_2_1"><span class="lab"><font color="Green" title="E16">A101</font></span></a>, <a class="txt" href="#E7:16_2_1"><span class="lab"><font color="Green" title="E17">A102</font></span></a>, <a class="txt" href="#E8:16_2_1"><span class="lab"><font color="Green" title="E18">A103</font></span></a>, <a class="txt" href="#E9:16_2_1"><span class="lab"><font color="Green" title="E19">A104</font></span></a>, <a class="txt" href="#E10:16_2_1"><span class="lab"><font color="Green" title="E20">A105</font></span></a>, <a class="txt" href="#E11:16_2_1"><span class="lab"><font color="Green" title="E21">A106</font></span></a>, <a class="txt" href="#E12:16_2_1"><span class="lab"><font color="Green" title="E22">A107</font></span></a>, <a class="txt" href="#E13:16_2_1"><span class="lab"><font color="Green" title="E23">A108</font></span></a>, <a class="txt" href="#E14:16_2_1"><span class="lab"><font color="Green" title="E24">A109</font></span></a>, <a class="txt" href="#E15:16_2_1"><span class="lab"><font color="Green" title="E25">A110</font></span></a>, <a class="txt" href="#E16:16_2_1"><span class="lab"><font color="Green" title="E26">A111</font></span></a>, <a class="txt" href="#E17:16_2_1"><span class="lab"><font color="Green" title="E27">A112</font></span></a>, <a class="txt" href="#E6:16_2_1_1"><span class="lab"><font color="Green" title="E36">A121</font></span></a>, <a class="txt" href="#E5:16_2_1_1"><span class="lab"><font color="Green" title="E35">A120</font></span></a>, <a class="txt" href="#E4:16_2_1_1"><span class="lab"><font color="Green" title="E34">A119</font></span></a>, <a class="txt" href="#E3:16_2_1_1"><span class="lab"><font color="Green" title="E33">A118</font></span></a>, <a class="txt" href="#E2:16_2_1_1"><span class="lab"><font color="Green" title="E32">A117</font></span></a>, <a class="txt" href="#E1:16_2_1_1"><span class="lab"><font color="Green" title="E31">A116</font></span></a>, <a class="ref" href="aff_1.html#T38" onmouseover="rs('aff_1/T38')" onmouseout="rh()">AFF_1:38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:16_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_2_1"><span class="lab"><font color="Green" title="E13">A98</font></span></a>, <a class="txt" href="#E4:16_2_1"><span class="lab"><font color="Green" title="E14">A99</font></span></a>, <a class="txt" href="#E5:16_2_1"><span class="lab"><font color="Green" title="E15">A100</font></span></a>, <a class="ref" href="aff_1.html#T45" onmouseover="rs('aff_1/T45')" onmouseout="rh()">AFF_1:45</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/>


</div><span class="kw">end;</span></div><a NAME="E21:16_2_1"/><span class="lab"><font color="Green" title="E31">A122</font></span>: 
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b1</font>,<font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16_2_1"><span class="lab"><font color="Green" title="E29">A114</font></span></a>, <a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E22:16_2_1"/><span class="lab"><font color="Green" title="E32">A123</font></span>: 
<font color="Maroon" title="c9">b1</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T2" onmouseover="rs('aff_1/T2')" onmouseout="rh()">AFF_1:2</a></span>;<br/><a NAME="E23:16_2_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_2_1"><span class="lab"><font color="Green" title="E28">A113</font></span></a>, <a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</a></span>;<br/><a NAME="E24:16_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2"><span class="lab"><font color="Green" title="E10">A95</font></span></a>, <a class="txt" href="#E1:16_2_1"><span class="lab"><font color="Green" title="E11">A96</font></span></a>, <a class="txt" href="#E2:16_2_1"><span class="lab"><font color="Green" title="E12">A97</font></span></a>, <a class="txt" href="#E4:16_2_1"><span class="lab"><font color="Green" title="E14">A99</font></span></a>, <a class="txt" href="#E5:16_2_1"><span class="lab"><font color="Green" title="E15">A100</font></span></a>, <a class="txt" href="#E12:16_2_1"><span class="lab"><font color="Green" title="E22">A107</font></span></a>, <a class="txt" href="#E13:16_2_1"><span class="lab"><font color="Green" title="E23">A108</font></span></a>, <a class="txt" href="#E14:16_2_1"><span class="lab"><font color="Green" title="E24">A109</font></span></a>, <a class="txt" href="#E15:16_2_1"><span class="lab"><font color="Green" title="E25">A110</font></span></a>, <a class="txt" href="#E16:16_2_1"><span class="lab"><font color="Green" title="E26">A111</font></span></a>, <a class="txt" href="#E17:16_2_1"><span class="lab"><font color="Green" title="E27">A112</font></span></a>, <a class="txt" href="#E20:16_2_1"><span class="lab"><font color="Green" title="E30">A115</font></span></a>, <a class="txt" href="#E21:16_2_1"><span class="lab"><font color="Green" title="E31">A122</font></span></a>, <a class="txt" href="#E22:16_2_1"><span class="lab"><font color="Green" title="E32">A123</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E25:16_2_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c10">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" onmouseover="rs('aff_1/T4')" onmouseout="rh()">AFF_1:4</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:16_2"/>
<font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_2.html#D2" onmouseover="rs('aff_2/D2')" onmouseout="rh()">AFF_2:def 2</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:16"/>
( <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V2" title="AFF_2:attr.2">Pappian</a>  iff <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V6" title="CONMETR1:attr.6">satisfying_major_indirect_Scherungssatz</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E9">A1</font></span></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/>


</div>
