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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">a</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">G</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Olive" title="b1">V</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">G</font> &amp; ( for <font color="Olive" title="b5">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b3">F</font> holds <br/><font color="Olive" title="b4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Olive" title="b2">a</font> ) holds <br/> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Olive" title="b2">a</font></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">a</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">F</font>, <font color="Olive" title="b3">G</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">G</font> &amp; ( for <font color="Olive" title="b4">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">F</font> holds <br/><font color="Olive" title="b3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Olive" title="b1">a</font> ) holds <br/> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Olive" title="b1">a</font></span><br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</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">F</font>, <font color="Olive" title="b2">G</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">G</font> &amp; ( for <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c3">a</font> ) holds <br/> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c3">a</font></span><br/><b>let </b><font color="Maroon" title="c4">F</font>, <font color="Maroon" title="c5">G</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c3">a</font> ) implies  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c3">a</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green" title="E4">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font>
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green" title="E5">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c3">a</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> holds <br/><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c3">a</font></span><br/><b>let </b><font color="Maroon" title="c7">v</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="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> implies <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c3">a</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:3_1"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">G</font>
 <b>and </b><br/><a NAME="E2:3_1"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c3">a</font></span><br/><a NAME="E3:3_1"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="#E1:3_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E4:3_1"/><b>then </b>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="#E2:3_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>hence </b><a NAME="E5:3_1"/>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="#E3:3_1"><i><font color="Green" title="E8">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>
<b>hence </b><a NAME="E4:3"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="rmod_4.html#T1" target="_self" title="RMOD_4:th.1">Th1</a></i>; <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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