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<span class="kw">let </span><font color="Maroon" title="c1">AFP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><br/>  for <font color="Olive" title="b3">K</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><br/>  for <font color="Olive" title="b4">f</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b3">K</font> &amp;  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">K</font> &amp; (  for <font color="Olive" title="b5">x</font>, <font color="Olive" title="b6">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> holds <br/> ( <font color="Olive" title="b4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">y</font> iff ( ( <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">K</font> &amp; <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">y</font> ) or (  not <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">K</font> &amp;  ex <font color="Olive" title="b7">p</font>, <font color="Olive" title="b8">p9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( <font color="Olive" title="b7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">K</font> &amp; <font color="Olive" title="b8">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">K</font> &amp; <font color="Olive" title="b7">p</font>,<font color="Olive" title="b1">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">p9</font>,<font color="Olive" title="b5">x</font> &amp; <font color="Olive" title="b7">p</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">p9</font>,<font color="Olive" title="b6">y</font> &amp; <font color="Olive" title="b5">x</font>,<font color="Olive" title="b6">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b3">K</font> ) ) ) ) ) holds <br/><font color="Olive" title="b4">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b1">K</font> &amp;  not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; (  for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> holds <br/> ( <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> iff ( ( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> ) or (  not <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp;  ex <font color="Olive" title="b5">p</font>, <font color="Olive" title="b6">p9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( <font color="Olive" title="b5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b6">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b5">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">p9</font>,<font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b5">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">p9</font>,<font color="Olive" title="b4">y</font> &amp; <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Olive" title="b1">K</font> ) ) ) ) ) holds <br/><font color="Olive" title="b2">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">K</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font> &amp;  not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; (  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> iff ( ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> ) or (  not <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp;  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">p9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( <font color="Olive" title="b4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b5">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b4">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">p9</font>,<font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b4">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">p9</font>,<font color="Olive" title="b3">y</font> &amp; <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font> ) ) ) ) ) holds <br/><font color="Olive" title="b1">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font> &amp;  not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; (  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> holds <br/> ( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> iff ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> ) or (  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp;  ex <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">p9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b4">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">p9</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">p9</font>,<font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font> ) ) ) ) ) implies <font color="Maroon" title="c5">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">and </span><br/><a NAME="E2:34"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
 not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">and </span><br/><a NAME="E3:34"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> holds <br/> ( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> iff ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> ) or (  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp;  ex <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">p9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b4">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">p9</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">p9</font>,<font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font> ) ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a> </span><br/>

<a NAME="E4:34"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon" title="c4">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E5:34"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
 not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="aff_1.html#T35" title="AFF_1:th.35">AFF_1:35</a></span>;<br/>
<a NAME="E6:34"/>
 for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  holds <br/><font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  implies <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c6">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">B9</font> = <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>;<br/>
<div><span class="lab"><font color="Green" title="E41">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34_1_1"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">and </span><br/><a NAME="E2:34_1_1"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:34_1_1_1"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c9">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E3:34_1_1_1"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T1" title="TRANSGEO:th.1">TRANSGEO:1</a></span>;<br/><a NAME="E4:34_1_1_1"/>
( ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> ) or (  not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp;  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b2">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">p9</font>,<font color="Maroon" title="c9">y</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">p9</font>,<font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E3:34_1_1_1"><span class="lab"><font color="Green" title="E45">A11</font></span></a></span>;<br/><a NAME="E5:34_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E2:34_1_1"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E1:34_1_1_1"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="aff_1.html#T34" title="AFF_1:th.34">AFF_1:34</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/><a NAME="E6:34_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a></span>;<br/><a NAME="E7:34_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E1:34_1_1_1"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/><span class="kw">hence </span><a NAME="E8:34_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_1_1"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="transgeo.html#T87" title="TRANSGEO:th.87">TRANSGEO:87</a></span>; <a class="txt" 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="E4:34_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A12</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T2" title="SUBSET_1:th.2">SUBSET_1:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:34_1_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E3:34_1_1_2"/><span class="lab"><font color="Green" title="E45">A13</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">and </span><br/><a NAME="E4:34_1_1_2"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T87" title="TRANSGEO:th.87">TRANSGEO:87</a></span>;<br/><a NAME="E5:34_1_1_2"/>
 not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E2:34_1_1"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E3:34_1_1_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/><a NAME="E6:34_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b2">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">p9</font>,<font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">p9</font>,<font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c8">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E4:34_1_1_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a></span>;<br/><a NAME="E7:34_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c8">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a></span>;<br/><span class="kw">hence </span><a NAME="E8:34_1_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E3:34_1_1_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>; <a class="txt" 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="E6:34_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T2" title="SUBSET_1:th.2">SUBSET_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:34_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E4:34_1_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" 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="E42">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:34_1_2"/>
<font color="Maroon" title="c4">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">M</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E3:34_1_2"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">M</font>
 <span class="kw">and </span><br/><a NAME="E4:34_1_2"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c4">K</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c8">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/><a NAME="E5:34_1_2"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
<font color="Maroon" title="c8">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1_2"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">A9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E7:34_1_2"/><span class="lab"><font color="Green" title="E46">A19</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A9</font>
 <span class="kw">and </span><br/><a NAME="E8:34_1_2"/><span class="lab"><font color="Green" title="E47">A20</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c9">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/><a NAME="E9:34_1_2"/><span class="lab"><font color="Green" title="E48">A21</font></span>: 
<font color="Maroon" title="c9">A9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_1_2"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E10:34_1_2"/><span class="lab"><font color="Green" title="E49">A22</font></span>: 
 not <font color="Maroon" title="c6">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">and </span><br/><a NAME="E11:34_1_2"/><span class="lab"><font color="Green" title="E50">A23</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c10">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E13:34_1_2"/><span class="lab"><font color="Green" title="E51">A24</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">and </span><br/><a NAME="E14:34_1_2"/><span class="lab"><font color="Green" title="E52">A25</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E10:34_1_2"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/><a NAME="E15:34_1_2"/>
 not <font color="Maroon" title="c9">A9</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_1_2"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E8:34_1_2"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E17:34_1_2"/><span class="lab"><font color="Green" title="E53">A26</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A9</font>
 <span class="kw">and </span><br/><a NAME="E18:34_1_2"/><span class="lab"><font color="Green" title="E54">A27</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E9:34_1_2"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">C9</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font>);<br/><a NAME="E19:34_1_2"/><span class="lab"><font color="Green" title="E55">A28</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E20:34_1_2"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c8">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:34_1_2"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="aff_1.html#T43" title="AFF_1:th.43">AFF_1:43</a></span>;<br/><a NAME="E21:34_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_2"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E5:34_1_2"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="ref" href="aff_1.html#T23" title="AFF_1:th.23">AFF_1:23</a></span>;<br/><a NAME="E22:34_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E3:34_1_2"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E4:34_1_2"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E18:34_1_2"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/><a NAME="E23:34_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E25:34_1_2"/><span class="lab"><font color="Green" title="E56">A29</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font>
 <span class="kw">and </span><br/><a NAME="E26:34_1_2"/><span class="lab"><font color="Green" title="E57">A30</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c13">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/><a NAME="E27:34_1_2"/><span class="lab"><font color="Green" title="E58">A31</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E28:34_1_2"/><span class="lab"><font color="Green" title="E59">A32</font></span>: 
<font color="Maroon" title="c13">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_1_2"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font> implies <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:34_1_2_1"/><span class="lab"><font color="Green" title="E60">A33</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font></span><br/><div><span class="lab"><font color="Green" title="E61">A34</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:34_1_2_1_1"/><span class="lab"><font color="Green" title="E62">A35</font></span>: 
<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c14">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_1_2"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E27:34_1_2"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="txt" href="#E25:34_1_2"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E26:34_1_2"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="txt" href="#E1:34_1_2_1"><span class="lab"><font color="Green" title="E60">A33</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E2:34_1_2_1_1"/>
<font color="Maroon" title="c4">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E4:34_1_2_1_1"/><span class="lab"><font color="Green" title="E63">A36</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N</font>
 <span class="kw">and </span><br/><a NAME="E5:34_1_2_1_1"/><span class="lab"><font color="Green" title="E64">A37</font></span>: 
<font color="Maroon" title="c4">K</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c15">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/><a NAME="E6:34_1_2_1_1"/><span class="lab"><font color="Green" title="E65">A38</font></span>: 
<font color="Maroon" title="c15">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_1_2_1_1"><span class="lab"><font color="Green" title="E64">A37</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><a NAME="E7:34_1_2_1_1"/>
 not <font color="Maroon" title="c15">N</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_1_2"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E5:34_1_2_1_1"><span class="lab"><font color="Green" title="E64">A37</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E9:34_1_2_1_1"/><span class="lab"><font color="Green" title="E66">A39</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N</font>
 <span class="kw">and </span><br/><a NAME="E10:34_1_2_1_1"/><span class="lab"><font color="Green" title="E67">A40</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E6:34_1_2_1_1"><span class="lab"><font color="Green" title="E65">A38</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/><a NAME="E11:34_1_2_1_1"/><span class="lab"><font color="Green" title="E68">A41</font></span>: 
<font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c14">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1_2_1_1"><span class="lab"><font color="Green" title="E63">A36</font></span></a>, <a class="txt" href="#E5:34_1_2_1_1"><span class="lab"><font color="Green" title="E64">A37</font></span></a>, <a class="txt" href="#E9:34_1_2_1_1"><span class="lab"><font color="Green" title="E66">A39</font></span></a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a></span>;<br/><span class="kw">assume </span><a NAME="E12:34_1_2_1_1"/><span class="lab"><font color="Green" title="E69">A42</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font></span><br/><a NAME="E13:34_1_2_1_1"/><span class="lab"><font color="Green" title="E70">A43</font></span>: 
 not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_1_2_1_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:34_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1_2_1_1"><span class="lab"><font color="Green" title="E63">A36</font></span></a>, <a class="txt" href="#E5:34_1_2_1_1"><span class="lab"><font color="Green" title="E64">A37</font></span></a>, <a class="txt" href="#E9:34_1_2_1_1"><span class="lab"><font color="Green" title="E66">A39</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<a NAME="E3:34_1_2_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E14:34_1_2"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E25:34_1_2"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E26:34_1_2"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="txt" href="#E28:34_1_2"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="txt" href="#E1:34_1_2_1"><span class="lab"><font color="Green" title="E60">A33</font></span></a>, <a class="txt" href="#E12:34_1_2_1_1"><span class="lab"><font color="Green" title="E69">A42</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:34_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E18:34_1_2"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E19:34_1_2"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E27:34_1_2"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>; <a class="txt" 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="E14:34_1_2_1_1"/>
<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_1_2"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E7:34_1_2"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E8:34_1_2"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="txt" href="#E17:34_1_2"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E10:34_1_2_1_1"><span class="lab"><font color="Green" title="E67">A40</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E15:34_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E14:34_1_2"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E18:34_1_2"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E13:34_1_2_1_1"><span class="lab"><font color="Green" title="E70">A43</font></span></a>, <a class="txt" href="#E1:34_1_2_1_1"><span class="lab"><font color="Green" title="E62">A35</font></span></a>, <a class="txt" href="#E11:34_1_2_1_1"><span class="lab"><font color="Green" title="E68">A41</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:34_1_2_1_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:34_1_2_1_1"><span class="lab"><font color="Green" title="E67">A40</font></span></a>, <a class="ref" href="transgeo.html#T87" title="TRANSGEO:th.87">TRANSGEO:87</a></span>; <a class="txt" 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" onclick="hs2(this)" href="javascript:()" title="34_1_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:34_1_2_1_2"/><span class="lab"><font color="Green" title="E62">A44</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font></span><br/><a NAME="E2:34_1_2_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E14:34_1_2"><span class="lab"><font color="Green" title="E52">A25</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_1_2_1_2"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_1_2"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E1:34_1_2_1_2"><span class="lab"><font color="Green" title="E62">A44</font></span></a>, <a class="ref" href="transgeo.html#T87" title="TRANSGEO:th.87">TRANSGEO:87</a></span>; <a class="txt" 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="E4:34_1_2_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1_2_1"><span class="lab"><font color="Green" title="E61">A34</font></span></a></span>; <a class="txt" 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="E30:34_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A45</font></span>: 
<font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T2" title="SUBSET_1:th.2">SUBSET_1:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> implies <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font> )</span><br/><span class="kw">assume </span><a NAME="E1:34_1_2_2"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E3:34_1_2_2"/><span class="lab"><font color="Green" title="E61">A46</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">and </span><br/><a NAME="E4:34_1_2_2"/><span class="lab"><font color="Green" title="E62">A47</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T87" title="TRANSGEO:th.87">TRANSGEO:87</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:34_1_2_2_1"/>
<font color="Maroon" title="c4">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E3:34_1_2_2_1"/><span class="lab"><font color="Green" title="E63">A48</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N</font>
 <span class="kw">and </span><br/><a NAME="E4:34_1_2_2_1"/><span class="lab"><font color="Green" title="E64">A49</font></span>: 
<font color="Maroon" title="c4">K</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c16">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/><a NAME="E5:34_1_2_2_1"/><span class="lab"><font color="Green" title="E65">A50</font></span>: 
 not <font color="Maroon" title="c13">C</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c16">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_1_2_2_1_1"/>
<font color="Maroon" title="c13">C</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c16">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:34_1_2_2_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c16">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:34_1_2"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<a NAME="E3:34_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A51</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1_2_2_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<a NAME="E4:34_1_2_2_1_1"/>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E5:34_1_2_2_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:34_1_2"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E3:34_1_2_2_1_1"><span class="lab"><font color="Green" title="E66">A51</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:34_1_2_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>; <a class="txt" 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="E6:34_1_2_2_1"/>
<font color="Maroon" title="c16">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1_2_2_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E8:34_1_2_2_1"/><span class="lab"><font color="Green" title="E66">A52</font></span>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font>
 <span class="kw">and </span><br/><a NAME="E9:34_1_2_2_1"/><span class="lab"><font color="Green" title="E67">A53</font></span>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:34_1_2"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="txt" href="#E5:34_1_2_2_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/><a NAME="E10:34_1_2_2_1"/><span class="lab"><font color="Green" title="E68">A54</font></span>: 
<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">z</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_2_2_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E4:34_1_2_2_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="txt" href="#E9:34_1_2_2_1"><span class="lab"><font color="Green" title="E67">A53</font></span></a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a></span>;<br/><span class="kw">assume </span><a NAME="E11:34_1_2_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font></span><br/><a NAME="E12:34_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A55</font></span>: 
 not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E11:34_1_2"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E13:34_1_2"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E14:34_1_2"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E3:34_1_2_2"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><a NAME="E13:34_1_2_2_1"/><span class="lab"><font color="Green" title="E70">A56</font></span>: 
<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_1_2"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E7:34_1_2"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E8:34_1_2"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="txt" href="#E17:34_1_2"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:34_1_2_2"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><a NAME="E14:34_1_2_2_1"/>
<font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c17">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:34_1_2"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="txt" href="#E27:34_1_2"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="txt" href="#E25:34_1_2"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E26:34_1_2"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="txt" href="#E8:34_1_2_2_1"><span class="lab"><font color="Green" title="E66">A52</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E15:34_1_2_2_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E14:34_1_2"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E18:34_1_2"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E4:34_1_2_2"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E12:34_1_2_2_1"><span class="lab"><font color="Green" title="E69">A55</font></span></a>, <a class="txt" href="#E8:34_1_2_2_1"><span class="lab"><font color="Green" title="E66">A52</font></span></a>, <a class="txt" href="#E13:34_1_2_2_1"><span class="lab"><font color="Green" title="E70">A56</font></span></a>, <a class="txt" href="#E10:34_1_2_2_1"><span class="lab"><font color="Green" title="E68">A54</font></span></a></span>; <a class="txt" 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="E6:34_1_2_2"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E14:34_1_2"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E25:34_1_2"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E4:34_1_2_2"><span class="lab"><font color="Green" title="E62">A47</font></span></a></span>; <a class="txt" 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="E32:34_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T2" title="SUBSET_1:th.2">SUBSET_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E33:34_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:34_1_2"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="txt" href="#E30:34_1_2"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" 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" onclick="hs2(this)" href="javascript:()" title="34_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:34_1_3"/><span class="lab"><font color="Green" title="E43">A57</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:34_1_3_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font></span><br/><a NAME="E2:34_1_3_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T87" title="TRANSGEO:th.87">TRANSGEO:87</a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_1_3_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E1:34_1_3"><span class="lab"><font color="Green" title="E43">A57</font></span></a></span>; <a class="txt" 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="E3:34_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A58</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T2" title="SUBSET_1:th.2">SUBSET_1:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:34_1_3_2"/><span class="lab"><font color="Green" title="E45">A59</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font></span><br/><span class="kw">set </span><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>;<br/><a NAME="E2:34_1_3_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_3_2"><span class="lab"><font color="Green" title="E45">A59</font></span></a>, <a class="ref" href="transgeo.html#T86" title="TRANSGEO:th.86">TRANSGEO:86</a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_1_3_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E1:34_1_3"><span class="lab"><font color="Green" title="E43">A57</font></span></a>, <a class="txt" href="#E1:34_1_3_2"><span class="lab"><font color="Green" title="E45">A59</font></span></a></span>; <a class="txt" 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="E5:34_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T2" title="SUBSET_1:th.2">SUBSET_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:34_1_3"/>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E3:34_1_3"><span class="lab"><font color="Green" title="E44">A58</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" 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="E5:34_1"/>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a></span>; <a class="txt" 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="E7:34"/>
<font color="Maroon" title="c5">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T92" title="TRANSGEO:th.92">TRANSGEO:92</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
