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<b>let </b><font color="Maroon">K</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V7">commutative</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>;<br/>





<b>reconsider </b><font color="Maroon">r</font> =  <a href="fvsum_1.html#K11" target="_self">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> ;<br/>
<b>reconsider </b><font color="Maroon">s</font> =  <a href="fvsum_1.html#K11" target="_self">mlt</a> <font color="Maroon">q</font>,<font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> ;<br/>
<a NAME="E3:83"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T85">FINSEQ_2:85</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> =  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<a NAME="E5:83"/><i><font color="Green">E55</font></i>: 
(  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font> &amp;  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:83"/>
(  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp;  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E7:83"/><b>then </b>
(  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E8:83"/><b>then </b><i><font color="Green">E81</font></i>: 
(  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font> &amp;  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:83_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font>
 ;<br/><b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>, <font color="Maroon">d2</font> = <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i><a class="txt" href="fvsum_1.html#E17"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>thus </b><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T14" target="_self">Th14</a>, <a class="ref" href="fvsum_1.html#T15" target="_self">Th15</a>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<font color="Maroon">d1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d2</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="fvsum_1.html#T14" target="_self">Th14</a>, <a class="ref" href="fvsum_1.html#T15" target="_self">Th15</a>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:83"/>
 <a href="fvsum_1.html#K11" target="_self">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K11" target="_self">mlt</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


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