<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</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">V</font> being  <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">v2</font> holds <br/>for <font color="Olive" title="b4">l</font> being   <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> holds   <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v2</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</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">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> holds <br/>for <font color="Olive" title="b3">l</font> being   <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> holds   <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v1</font></span>)</span></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">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v2</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">v2</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v2</font> implies for <font color="Olive" title="b1">l</font> being   <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> holds   <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v2</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">l</font> being   <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> holds   <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c5">l</font> be    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:39"/><i><font color="Green" title="E26">A2</font></i>: 
 <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="rmod_4.html#D7" target="_self" title="RMOD_4:def.7">Def7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1"/>
(  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 </span><b>by </b><i><a class="txt" href="rmod_4.html#E2:39"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T42" title="ZFMISC_1:th.42">ZFMISC_1:42</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1"/>
 <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:39_1_1_1"/><b>then </b><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c5">l</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="rmod_4.html#D6" target="_self" title="RMOD_4:def.6">Def6</a></i>;<br/><b>hence </b> <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c5">l</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E49"><i><font color="Green" title="E22">Lm3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_1.html#T37" title="MOD_1:th.37">MOD_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mod_1.html#T37" title="MOD_1:th.37">MOD_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E2:39_1_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="rmod_4.html#T22" target="_self" title="RMOD_4:th.22">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E2:39_1_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="rmod_4.html#T22" target="_self" title="RMOD_4:th.22">Th22</a></i>
;<br/> <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="39_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_2"/><i><font color="Green" title="E27">A4</font></i>: 
 <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">L</font> = <font color="Maroon" title="c5">l</font> as    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <b>by </b><i><a class="ref" href="rmod_4.html#D7" target="_self" title="RMOD_4:def.7">Def7</a></i>;<br/><a NAME="E3:39_1_1_2"/><i><font color="Green" title="E28">A5</font></i>: 
not <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c5">l</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="rmod_4.html#E1:39_1_1_2"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>thus </b> <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c5">l</font> = 
 <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c6">L</font>

<br/>.= 
<font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#T43" target="_self" title="RMOD_4:th.43">Th43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mod_1.html#T37" title="MOD_1:th.37">MOD_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E3:39_1_1_2"><i><font color="Green" title="E28">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="39_1_1_3"><b>suppose </b></a><a NAME="E1:39_1_1_3"/><i><font color="Green" title="E27">A6</font></i>: 
 <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">L</font> = <font color="Maroon" title="c5">l</font> as    <a href="rmod_4.html#M2" title="RMOD_4:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <b>by </b><i><a class="ref" href="rmod_4.html#D7" target="_self" title="RMOD_4:def.7">Def7</a></i>;<br/><a NAME="E3:39_1_1_3"/><i><font color="Green" title="E28">A7</font></i>: 
not <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c5">l</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="rmod_4.html#E1:39_1_1_3"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>thus </b> <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c5">l</font> = 
 <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c6">L</font>

<br/>.= 
<font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#T43" target="_self" title="RMOD_4:th.43">Th43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</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="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mod_1.html#T37" title="MOD_1:th.37">MOD_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E3:39_1_1_3"><i><font color="Green" title="E28">A7</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="39_1_1_4"><b>suppose </b></a><a NAME="E1:39_1_1_4"/>
 <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">F</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:39_1_1_4"/><i><font color="Green" title="E27">A8</font></i>: 
( <font color="Maroon" title="c6">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 <b>and </b><br/><a NAME="E4:39_1_1_4"/><i><font color="Green" title="E28">A9</font></i>: 
 <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="rmod_4.html#D9" target="_self" title="RMOD_4:def.9">Def9</a></i>;<br/><a NAME="E5:39_1_1_4"/>
( <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> or <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c3">v1</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="rmod_4.html#E3:39_1_1_4"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="finseq_3.html#T108" title="FINSEQ_3:th.108">FINSEQ_3:108</a></i>;<br/><a NAME="E6:39_1_1_4"/><b>then </b>
( <font color="Maroon" title="c5">l</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> or <font color="Maroon" title="c5">l</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="rmod_4.html#T35" target="_self" title="RMOD_4:th.35">Th35</a></i>;<br/><b>hence </b><a NAME="E7:39_1_1_4"/>
 <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E4:39_1_1_4"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="rlvect_1.html#T62" title="RLVECT_1:th.62">RLVECT_1:62</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="E4:39"/>
 <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v2</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
