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<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">T</font>, <font color="Olive" title="b3">S</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b2">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">S</font> holds <br/> <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Olive" title="b2">T</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Olive" title="b3">S</font></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">T</font>, <font color="Olive" title="b2">S</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">S</font> holds <br/> <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Olive" title="b2">S</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">T</font>, <font color="Maroon" title="c4">S</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">S</font> implies  <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c4">S</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c3">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c4">S</font></span>)</span></span><br/>

<b>consider </b><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><b> such that </b><br/><a NAME="E3:12"/><i><font color="Green" title="E8">A2</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">S</font>
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green" title="E9">A3</font></i>: 
( <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="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c5">F</font> )
 <b>by </b><i><a class="ref" href="rmod_4.html#D3" target="_self" title="RMOD_4:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">G</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="E6:12"/><i><font color="Green" title="E10">A4</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c6">G</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>and </b><br/><a NAME="E7:12"/><i><font color="Green" title="E11">A5</font></i>: 
 <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c6">G</font>
 <b>by </b><i><a class="ref" href="rmod_4.html#D3" target="_self" title="RMOD_4:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">H</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="E9:12"/><i><font color="Green" title="E12">A6</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c7">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>and </b><br/><a NAME="E10:12"/><i><font color="Green" title="E13">A7</font></i>: 
 <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="ref" href="rmod_4.html#D3" target="_self" title="RMOD_4:def.3">Def3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">I</font> = <font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">H</font>;<br/>
<a NAME="E11:12"/><i><font color="Green" title="E14">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="txt" href="#E3:12"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="#E6:12"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="#E9:12"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E12:12"/>
<font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="#E1:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E6:12"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="#E9:12"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/>
<b>hence </b> <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">S</font></span>)</span> = 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">H</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:12"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="#E11:12"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="rlvect_1.html#T59" title="RLVECT_1:th.59">RLVECT_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c4">S</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:12"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="#E10:12"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</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/>


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