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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</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">y</font> holds <br/><span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Olive" title="b1">x</font></span>)</span>,<font color="Olive" title="b2">y</font></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x</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="c3">y</font> holds <br/><span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">y</font></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span><br/><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</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="c2">x</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="c3">y</font> implies <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">y</font></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> )</span><br/>





<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E9">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x</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="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">y</font></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span><br/>

<a NAME="E2:12"/><i><font color="Green" title="E10">A3</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix14.html#T8" target="_self" title="MATRIX14:th.8">BB243</a>, <a class="txt" href="#E1:12"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E3:12"/><i><font color="Green" title="E11">A4</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="fvsum_1.html#T92" title="FVSUM_1:th.92">FVSUM_1:92</a></i>;<br/>
<a NAME="E4:12"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
 
<b>by </b><i><a class="txt" href="#E3:12"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="fvsum_1.html#D10" title="FVSUM_1:def.10">FVSUM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:12"/>
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c2">x</font></span>)</span>,<font color="Maroon" title="c3">y</font></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
 <b>by </b><i><a class="ref" href="fvsum_1.html#D10" title="FVSUM_1:def.10">FVSUM_1: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/>


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