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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">L1</font>, <font color="Olive" title="b2">L2</font> being  <a href="circled1.html#NM1" title="CIRCLED1:NM.1">circled_Combination</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b4">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Olive" title="b1">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Olive" title="b2">L2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Olive" title="b1">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Olive" title="b2">L2</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">L1</font>, <font color="Maroon" title="c3">L2</font> be   <a href="circled1.html#NM1" title="CIRCLED1:NM.1">circled_Combination</a> of <font color="Maroon" title="c1">V</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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:36"/>
<font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green" title="E17">A1</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T133" title="XREAL_1:th.133">XREAL_1:133</a></i>;<br/>
<a NAME="E3:36"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
(  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c2">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> &amp;  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T65" title="RLVECT_2:th.65">RLVECT_2:65</a></i>;<br/>
<a NAME="E4:36"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN: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="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN: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">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="circled1.html#E1:36_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN: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="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:36_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 <b>by </b><i><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:36_1_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E5:36_1_1_1"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c2">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="circled1.html#E3:36"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="circled1.html#E1:36_1_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="circled1.html#E4:36_1_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="circled1.html#T17" target="_self" title="CIRCLED1:th.17">Th17</a></i>;<br/><a NAME="E6:36_1_1_1"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_1_1"/>
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">L2</font> or  not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">L2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1_1_1_1"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:36_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon" title="c3">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="circled1.html#T17" target="_self" title="CIRCLED1:th.17">Th17</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span><b>by </b><i><a class="txt" href="circled1.html#E2:36"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:36_1_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="circled1.html#E5:36_1_1_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="circled1.html#E2:36_1_1_1_1_1_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/><a NAME="E3:36_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E25">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="rlvect_2.html#D13" title="RLVECT_2:def.13">RLVECT_2:def 13</a></i>;<br/><a NAME="E4:36_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E5:36_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="circled1.html#E3:36_1_1_1_1_1_1_1_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="rlvect_2.html#D12" title="RLVECT_2:def.12">RLVECT_2:def 12</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28" title="RLVECT_2:th.28">RLVECT_2:28</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="36_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:36_1_1_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="circled1.html#E3:36"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="circled1.html#E1:36_1_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="circled1.html#E4:36_1_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="circled1.html#E1:36_1_1_1_1_1_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="circled1.html#T17" target="_self" title="CIRCLED1:th.17">Th17</a>, <a class="ref" href="xreal_1.html#T134" title="XREAL_1:th.134">XREAL_1:134</a></i>;<br/><a NAME="E3:36_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E25">A10</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="rlvect_2.html#D13" title="RLVECT_2:def.13">RLVECT_2:def 13</a></i>;<br/><a NAME="E4:36_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/><a NAME="E5:36_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="circled1.html#E3:36_1_1_1_1_1_1_1_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="rlvect_2.html#D12" title="RLVECT_2:def.12">RLVECT_2:def 12</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28" title="RLVECT_2:th.28">RLVECT_2:28</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="36_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_1_1_1_2"/>
not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:36_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28" title="RLVECT_2:th.28">RLVECT_2:28</a></i>;<br/><a NAME="E3:36_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:36_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="rlvect_2.html#D13" title="RLVECT_2:def.13">RLVECT_2:def 13</a></i>;<br/><a NAME="E5:36_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 ;<br/><a NAME="E6:36_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="circled1.html#E4:36_1_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="rlvect_2.html#D12" title="RLVECT_2:def.12">RLVECT_2:def 12</a></i>;<br/><b>hence </b><a NAME="E7:36_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28" title="RLVECT_2:th.28">RLVECT_2:28</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><b>hence </b><a NAME="E7:36_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="circled1.html#E4:36_1_1_1"><i><font color="Green" title="E21">A5</font></i></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="36_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_2"/><i><font color="Green" title="E20">A11</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN: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">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:36_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 <b>by </b><i><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:36_1_1_2"/><i><font color="Green" title="E21">A12</font></i>: 
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E5:36_1_1_2"/><i><font color="Green" title="E22">A13</font></i>: 
<font color="Maroon" title="c3">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="circled1.html#E3:36"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="circled1.html#E1:36_1_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="circled1.html#E4:36_1_1_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="circled1.html#T17" target="_self" title="CIRCLED1:th.17">Th17</a></i>;<br/><a NAME="E6:36_1_1_2"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_2_1_1"/>
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c2">L1</font> or  not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c2">L1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_2_1_1_1"/><i><font color="Green" title="E23">A14</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c2">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E24">A15</font></i>: 
<font color="Maroon" title="c2">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="circled1.html#T17" target="_self" title="CIRCLED1:th.17">Th17</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_2_1_1_1_1"/>
( ( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span><b>by </b><i><a class="txt" href="circled1.html#E2:36"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="circled1.html#E5:36_1_1_2"><i><font color="Green" title="E22">A13</font></i></a>, <a class="txt" href="circled1.html#E2:36_1_1_2_1_1_1"><i><font color="Green" title="E24">A15</font></i></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/><a NAME="E3:36_1_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E25">A16</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="rlvect_2.html#D13" title="RLVECT_2:def.13">RLVECT_2:def 13</a></i>;<br/><a NAME="E4:36_1_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E5:36_1_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="circled1.html#E3:36_1_1_2_1_1_1_1_1"><i><font color="Green" title="E25">A16</font></i></a>, <a class="ref" href="rlvect_2.html#D12" title="RLVECT_2:def.12">RLVECT_2:def 12</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28" title="RLVECT_2:th.28">RLVECT_2:28</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="36_1_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_2_1_1_1_1_2"/>
( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="circled1.html#E3:36"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="circled1.html#E1:36_1_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="circled1.html#E4:36_1_1_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="circled1.html#E1:36_1_1_2_1_1_1"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="circled1.html#T17" target="_self" title="CIRCLED1:th.17">Th17</a>, <a class="ref" href="xreal_1.html#T134" title="XREAL_1:th.134">XREAL_1:134</a></i>;<br/><a NAME="E3:36_1_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E25">A17</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="rlvect_2.html#D13" title="RLVECT_2:def.13">RLVECT_2:def 13</a></i>;<br/><a NAME="E4:36_1_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/><a NAME="E5:36_1_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="circled1.html#E3:36_1_1_2_1_1_1_1_2"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="rlvect_2.html#D12" title="RLVECT_2:def.12">RLVECT_2:def 12</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28" title="RLVECT_2:th.28">RLVECT_2:28</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="36_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_2_1_1_2"/>
not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c2">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28" title="RLVECT_2:th.28">RLVECT_2:28</a></i>;<br/><a NAME="E3:36_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:36_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="rlvect_2.html#D13" title="RLVECT_2:def.13">RLVECT_2:def 13</a></i>;<br/><a NAME="E5:36_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 ;<br/><a NAME="E6:36_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="circled1.html#E4:36_1_1_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="rlvect_2.html#D12" title="RLVECT_2:def.12">RLVECT_2:def 12</a></i>;<br/><b>hence </b><a NAME="E7:36_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28" title="RLVECT_2:th.28">RLVECT_2:28</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><b>hence </b><a NAME="E7:36_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="circled1.html#E4:36_1_1_2"><i><font color="Green" title="E21">A12</font></i></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>
<a NAME="E5:36"/><b>then </b><i><font color="Green" title="E19">A18</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:36"/>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T58" title="RLVECT_2:th.58">RLVECT_2:58</a></i>;<br/>
<b>hence </b><a NAME="E7:36"/>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c2">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c3">L2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="circled1.html#E5:36"><i><font color="Green" title="E19">A18</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 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>
