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<span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="symsp_1.html#NM1" title="SYMSP_1:NM.1">SymSp</a> of <font color="Maroon" title="c4">F</font><br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">S</font>  st <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> &amp;  not <font color="Olive" title="b2">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/>(  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> iff <font color="Olive" title="b4">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">S</font> be   <a href="symsp_1.html#NM1" title="SYMSP_1:NM.1">SymSp</a> of <font color="Maroon" title="c4">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">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="c5">S</font>  st <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> &amp;  not <font color="Olive" title="b1">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/>(  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> iff <font color="Olive" title="b3">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</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"> ( <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> &amp;  not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  implies (  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> iff <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) )</span><br/>











<span class="kw">set </span><font color="Maroon" title="c10">0F</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
 not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</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"> (  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> iff <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span><br/>

<a NAME="E3:47"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="symsp_1.html#T1" target="_self" onmouseover="rs('symsp_1/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="symsp_1.html#T2" target="_self" onmouseover="rs('symsp_1/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E4:47"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
(  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> implies <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:47_1"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</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="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> </span><br/>

<div><span class="lab"><font color="Green" title="E46">A6</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_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"> (  ex <font color="Olive" title="b1">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>(  not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) implies <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c11">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font><span class="kw"> such that </span><a NAME="E2:47_1_1"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
 not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">and </span><br/><a NAME="E3:47_1_1"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
 not <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> </span><br/><div><span class="lab"><font color="Green" title="E49">A9</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_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">  not  <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:47_1_1_1"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
 <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</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="E2:47_1_1_1"/>
 not <font color="Maroon" title="c11">p</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="symsp_1.html#T2" target="_self" onmouseover="rs('symsp_1/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E3:47_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="symsp_1.html#T23" target="_self" onmouseover="rs('symsp_1/T23')" onmouseout="rh()">Th23</a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="symsp_1.html#T2" target="_self" onmouseover="rs('symsp_1/T2')" onmouseout="rh()">Th2</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="E5:47_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">p</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">x</font>)</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E3:47_1_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="symsp_1.html#D3" target="_self" onmouseover="rs('symsp_1/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E6:47_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">p</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> or  <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T12" onmouseover="rs('vectsp_1/T12')" onmouseout="rh()">VECTSP_1:12</a></span>;<br/><a NAME="E7:47_1_1"/><span class="kw">then </span>
(  <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> or  <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> or  <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T12" onmouseover="rs('vectsp_1/T12')" onmouseout="rh()">VECTSP_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E8:47_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E3:47_1_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E4:47_1_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="symsp_1.html#T23" target="_self" onmouseover="rs('symsp_1/T23')" onmouseout="rh()">Th23</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="47_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"> ( (  for <font color="Olive" title="b1">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  holds <br/>( <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  or <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) ) implies <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:47_1_2"/>
 for <font color="Olive" title="b1">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  holds <br/>( <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  or <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> </span><br/><a NAME="E2:47_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="symsp_1.html#T9" target="_self" onmouseover="rs('symsp_1/T9')" onmouseout="rh()">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="symsp_1.html#T1" target="_self" onmouseover="rs('symsp_1/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="symsp_1.html#T2" target="_self" onmouseover="rs('symsp_1/T2')" onmouseout="rh()">Th2</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="E4:47_1"/>
<font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1"><span class="lab"><font color="Green" title="E46">A6</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="E5:47"/>
( <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  implies  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:47_2"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</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">  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font></span><br/>

<div><span class="lab"><font color="Green" title="E46">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_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="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font> implies  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:47_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</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="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font><span class="kw"> such that </span><br/><a NAME="E3:47_2_1"/><span class="lab"><font color="Green" title="E47">A13</font></span>: 
 not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">and </span><br/><a NAME="E4:47_2_1"/><span class="lab"><font color="Green" title="E48">A14</font></span>: 
 not <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="symsp_1.html#T9" target="_self" onmouseover="rs('symsp_1/T9')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E5:47_2_1"/>
 <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="txt" href="#E4:47_2_1"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="ref" href="symsp_1.html#T23" target="_self" onmouseover="rs('symsp_1/T23')" onmouseout="rh()">Th23</a></span>;<br/><a NAME="E6:47_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c11">p</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">x</font>)</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c9">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font>
 ;<br/><span class="kw">hence </span><a NAME="E7:47_2_1"/>
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:47_2_1"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="txt" href="#E4:47_2_1"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="ref" href="symsp_1.html#D3" target="_self" onmouseover="rs('symsp_1/D3')" onmouseout="rh()">Def3</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="47_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="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font> implies  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:47_2_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</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="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font></span><br/><a NAME="E2:47_2_2"/><span class="kw">then </span>
 for <font color="Olive" title="b1">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  holds <br/>( <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  or <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="symsp_1.html#T1" target="_self" onmouseover="rs('symsp_1/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="symsp_1.html#T2" target="_self" onmouseover="rs('symsp_1/T2')" onmouseout="rh()">Th2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_2_2"/>
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="symsp_1.html#D3" target="_self" onmouseover="rs('symsp_1/D3')" onmouseout="rh()">Def3</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="E4:47_2"/>
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_2"><span class="lab"><font color="Green" title="E46">A12</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="E6:47"/>
(  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> iff <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E44">A4</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>
