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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">V</font> being   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a><br/>  for <font color="Olive" title="b2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font>  st  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">A</font> holds <br/> <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Olive" title="b2">A</font>,<font color="Maroon" title="c1">x</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">V</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/> <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">x</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</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="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies  <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">Ax</font> =  <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>);<br/>
<span class="kw">assume </span><a NAME="E1:47"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A</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="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><br/>

<div><span class="lab"><font color="Green" title="E38">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:47_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</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"> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><a NAME="E2:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#D3" target="_self" onmouseover="rs('rlaffin3/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E3:47_1"/>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" onmouseover="rs('rlvect_2/D6')" onmouseout="rh()">RLVECT_2:def 6</a></span>;<br/><a NAME="E4:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A4</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1_1"/>
( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font> or  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</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="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:47_1_1_1"/>
<span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E4:47_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="rlvect_2.html#T19" onmouseover="rs('rlvect_2/T19')" onmouseout="rh()">RLVECT_2:19</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" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_2"/>
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</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="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><div class="add"><a NAME="E2:47_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">A</font></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E3:47_1_1_2"/>
<span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E3:47"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:47"/>
 <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">x</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="funcop_1.html#T11" onmouseover="rs('funcop_1/T11')" onmouseout="rh()">FUNCOP_1:11</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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