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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b3">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font> or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>suppose </b></a><a NAME="E1:10_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:10_1_1"/><b>then </b><i><font color="Green" title="E7">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:10_1_1_1_1"/><b>then </b><i><font color="Green" title="E9">A4</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="txt" href="#E1:10_1_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="xboole_0.html#T1" title="XBOOLE_0:th.1">XBOOLE_0:1</a></i>;<br/><a NAME="E3:10_1_1_1_1"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="txt" href="#E1:10_1_1_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/><a NAME="E4:10_1_1_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">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> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="txt" href="#E2:10_1_1_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:10_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:10_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="#E3:10_1_1_1_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_2"/><i><font color="Green" title="E8">A6</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:10_1_1_1_2"/><b>then </b><i><font color="Green" title="E9">A7</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="txt" href="#E1:10_1_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="xboole_0.html#T1" title="XBOOLE_0:th.1">XBOOLE_0:1</a></i>;<br/><a NAME="E3:10_1_1_1_2"/><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">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> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="txt" href="#E1:10_1_1_1_2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/><a NAME="E4:10_1_1_1_2"/>
<font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="txt" href="#E2:10_1_1_1_2"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:10_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:10_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="#E3:10_1_1_1_2"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2"><b>suppose </b></a><a NAME="E1:10_1_2"/><i><font color="Green" title="E6">A9</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:10_1_2"/><b>then </b><i><font color="Green" title="E7">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">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> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_1"><b>suppose </b></a><a NAME="E1:10_1_2_1_1"/><i><font color="Green" title="E8">A11</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:10_1_2_1_1"/><b>then </b><i><font color="Green" title="E9">A12</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="txt" href="#E1:10_1_2"><i><font color="Green" title="E6">A9</font></i></a>, <a class="ref" href="xboole_0.html#T1" title="XBOOLE_0:th.1">XBOOLE_0:1</a></i>;<br/><a NAME="E3:10_1_2_1_1"/>
<font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="txt" href="#E1:10_1_2_1_1"><i><font color="Green" title="E8">A11</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:10_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:10_1_2"><i><font color="Green" title="E7">A10</font></i></a>, <a class="txt" href="#E2:10_1_2_1_1"><i><font color="Green" title="E9">A12</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a>, <a class="ref" href="bspace.html#T7" target="_self" title="BSPACE:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_2"><b>suppose </b></a><a NAME="E1:10_1_2_1_2"/><i><font color="Green" title="E8">A13</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:10_1_2_1_2"/><b>then </b><i><font color="Green" title="E9">A14</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="txt" href="#E1:10_1_2"><i><font color="Green" title="E6">A9</font></i></a>, <a class="ref" href="xboole_0.html#T1" title="XBOOLE_0:th.1">XBOOLE_0:1</a></i>;<br/><a NAME="E3:10_1_2_1_2"/><i><font color="Green" title="E10">A15</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">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> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="txt" href="#E1:10_1_2_1_2"><i><font color="Green" title="E8">A13</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/><a NAME="E4:10_1_2_1_2"/>
<font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">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> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="txt" href="#E2:10_1_2_1_2"><i><font color="Green" title="E9">A14</font></i></a>, <a class="ref" href="bspace.html#D3" target="_self" title="BSPACE:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:10_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:10_1_2"><i><font color="Green" title="E7">A10</font></i></a>, <a class="txt" href="#E3:10_1_2_1_2"><i><font color="Green" title="E10">A15</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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