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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> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">S</font>  st  not <font color="Olive" title="b2">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/>( <font color="Olive" title="b2">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  iff  <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">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><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> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  st  not <font color="Olive" title="b1">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/>( <font color="Olive" title="b1">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  iff  <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">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><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> 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"> (  not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  implies ( <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  iff  <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="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><br/>









<span class="kw">set </span><font color="Maroon" title="c9">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">set </span><font color="Maroon" title="c10">0V</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font>;<br/>
<div><span class="lab"><font color="Green" title="E30">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <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#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="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><span class="kw">that </span><br/><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E31">A2</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="E2:36_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<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#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="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><br/><a NAME="E3:36_1"/>
<font color="Maroon" title="c6">a</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="#E2:36_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</a></span>;<br/><a NAME="E4:36_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</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="rlvect_1.html#T12" onmouseover="rs('rlvect_1/T12')" onmouseout="rh()">RLVECT_1:12</a></span>;<br/><a NAME="E5:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c6">a</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="vectsp_1.html#T14" onmouseover="rs('vectsp_1/T14')" onmouseout="rh()">VECTSP_1:14</a></span>;<br/><a NAME="E6:36_1"/>
<font color="Maroon" title="c6">a</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="#E1:36_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="symsp_1.html#T14" target="_self" onmouseover="rs('symsp_1/T14')" onmouseout="rh()">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E7:36_1"/>
 <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="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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="symsp_1.html#T12" target="_self" onmouseover="rs('symsp_1/T12')" onmouseout="rh()">Th12</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="36_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  <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="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> implies <font color="Maroon" title="c6">a</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="E1:36_2"/>
(  not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  <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="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"> <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> </span><br/><a NAME="E2:36_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</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#T14" target="_self" onmouseover="rs('symsp_1/T14')" onmouseout="rh()">Th14</a></span>;<br/><a NAME="E3:36_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</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="vectsp_1.html#T14" onmouseover="rs('vectsp_1/T14')" onmouseout="rh()">VECTSP_1:14</a></span>;<br/><a NAME="E4:36_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</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="rlvect_1.html#T12" onmouseover="rs('rlvect_1/T12')" onmouseout="rh()">RLVECT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_2"/>
<font color="Maroon" title="c6">a</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="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:36"/>
(  not <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  implies ( <font color="Maroon" title="c6">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  iff  <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="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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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