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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">RightMod</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">v</font> being   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Olive" title="b1">V</font><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#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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> <font color="Olive" title="b2">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<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">v</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">RightMod</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">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#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds   <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">v</font> be   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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> <font color="Maroon" title="c3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<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">v</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span></span><br/>







<a NAME="E1:38"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_4.html#D5" target="_self" onmouseover="rs('rmod_4/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_1"/>
(  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">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="c4">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">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" onmouseover="rs('zfmisc_1/T33')" onmouseout="rh()">ZFMISC_1:33</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1"/>
 <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span></span><br/><div class="add"><a NAME="E2:38_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_4.html#D4" target="_self" onmouseover="rs('rmod_4/D4')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">hence </span> <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c4">l</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="rmod_4.html#E38"><span class="lab"><font color="Green" title="E23">Lm3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<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 class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_2.html#T32" onmouseover="rs('vectsp_2/T32')" onmouseout="rh()">VECTSP_2:32</a></span>
<br/>.= 
<font color="Maroon" title="c3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="rmod_4.html#T18" target="_self" onmouseover="rs('rmod_4/T18')" onmouseout="rh()">Th18</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_2"/>
 <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">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">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">F</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:38_1_1_2"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
( <font color="Maroon" title="c5">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">F</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">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">and </span><br/><a NAME="E4:38_1_1_2"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
 <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_4.html#D7" target="_self" onmouseover="rs('rmod_4/D7')" onmouseout="rh()">Def7</a></span>;<br/><a NAME="E5:38_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_2"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="finseq_3.html#T97" onmouseover="rs('finseq_3/T97')" onmouseout="rh()">FINSEQ_3:97</a></span>;<br/><a NAME="E6:38_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">l</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_4.html#T25" target="_self" onmouseover="rs('rmod_4/T25')" onmouseout="rh()">Th25</a></span>;<br/><span class="kw">hence </span><a NAME="E7:38_1_1_2"/>
 <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_1_2"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="rlvect_1.html#T44" onmouseover="rs('rlvect_1/T44')" onmouseout="rh()">RLVECT_1:44</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:38"/>
 <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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