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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>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</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> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">y</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">z</font> holds <br/> <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Olive" title="b3">z</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Olive" title="b3">z</font>,<font color="Olive" title="b1">x</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Olive" title="b3">z</font>,<font color="Olive" title="b2">y</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</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"> (  <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> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y</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="c4">z</font> implies  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c4">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c3">y</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E25">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> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y</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="c4">z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c4">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">x2</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c6">y2</font> = <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c7">z2</font> = <font color="Maroon" title="c4">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
 <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c4">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span> = 
 <a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c7">z2</font>
<b>by </b><i><a class="ref" href="fvsum_1.html#T77" title="FVSUM_1:th.77">FVSUM_1:77</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">z2</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_4.html#E1:55"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="matrix_4.html#T56" target="_self" title="MATRIX_4:th.56">Th56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c7">z2</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T77" title="FVSUM_1:th.77">FVSUM_1:77</a></i>
;<br/>
<b>hence </b><a NAME="E4:55"/>
 <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c4">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="fvsum_1.html#T77" title="FVSUM_1:th.77">FVSUM_1:77</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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