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<b>let </b><font color="Maroon">GF</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V12">VectSp-like</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">L1</font> be    <a href="vectsp_6.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">L2</font> be    <a href="vectsp_6.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>;<br/>







<b>consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E2:53"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E3:53"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span>
 <b>and </b><br/><a NAME="E4:53"/><i><font color="Green">E59</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:53"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">G</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E7:53"/><i><font color="Green">E61</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font>
 <b>and </b><br/><a NAME="E8:53"/><i><font color="Green">E70</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">L1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">H</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E10:53"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">H</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E11:53"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font>
 <b>and </b><br/><a NAME="E12:53"/><i><font color="Green">E73</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">L2</font>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">C1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E14:53"/><i><font color="Green">E79</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span>
 <b>and </b><br/><a NAME="E15:53"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</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">V</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E17:53"/><i><font color="Green">E81</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:53_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:53_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:53_1"/>
<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E2:53"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:53_1"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="vectsp_6.html#E2:53"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:53_1"/><b>then </b>
not <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:53_1"/>
<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E89</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_3.html#T10">VECTSP_3:10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>
;<br/>
<b>set </b><font color="Maroon">GG</font> = <font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>;<br/>
<i><font color="Green">E90</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_6.html#E2:53"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>set </b><font color="Maroon">C2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E22:53"/><i><font color="Green">E91</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span>
 <b>and </b><br/><a NAME="E23:53"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">q</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</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">V</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E25:53"/><i><font color="Green">E93</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:53_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:53_4"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:53_4"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:53_4"/>
<font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:53_4"/><b>then </b>
<font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:53_4"/><b>then </b>
not <font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:53_4"/>
<span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E96</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">q</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_3.html#T10">VECTSP_3:10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>
;<br/>
<b>set </b><font color="Maroon">HH</font> = <font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font>;<br/>
<b>set </b><font color="Maroon">h</font> = <font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>;<br/>
<i><font color="Green">E97</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">H</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">H</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>set </b><font color="Maroon">C3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E30:53"/><i><font color="Green">E98</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E31:53"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">r</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E33:53"/><i><font color="Green">E100</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:53_7"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:53_7"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:53_7"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:53_7"/>
<font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:53_7"/><b>then </b>
<font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:53_7"/><b>then </b>
not <font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:53_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E105</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_3.html#T10">VECTSP_3:10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>
;<br/>
<b>set </b><font color="Maroon">FF</font> = <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>;<br/>
<i><font color="Green">E106</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<a NAME="E37:53"/><i><font color="Green">E107</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_10"/>
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:53_10"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E4:53_10"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span> )
 
<b>by </b><i>, , <a class="txt" href="vectsp_6.html#E6:53"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:53_10"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:53"/><i><font color="Green">E109</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_11"/>
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:53_11"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span>;<br/>
<a NAME="E4:53_11"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span> )
 
<b>by </b><i>, , <a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:53_11"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:53"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_12"/>
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:53_12"/><b>then </b><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font></span>)</span>;<br/>
<a NAME="E4:53_12"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:53_12"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E40:53"/><b>then </b><i><font color="Green">E112</font></i>: 
( <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i>, , , <a class="ref" href="vectsp_6.html#D4" target="_self">Def4</a>, , <a class="txt" href="vectsp_6.html#E3:53"><i><font color="Green">E55</font></i></a>, <a class="txt" href="vectsp_6.html#E4:53"><i><font color="Green">E59</font></i></a>, <a class="txt" href="vectsp_6.html#E6:53"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E41:53"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E42:53"/><b>then </b><i><font color="Green">E113</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , , , , , <a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/>
<a NAME="E43:53"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E44:53"/><b>then </b>
(  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span> &amp;  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span> &amp;  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E45:53"/><b>then </b><i><font color="Green">E114</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T22" target="_self">Th22</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E46:53"/><b>then </b><i><font color="Green">E115</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#T65">FINSEQ_1:65</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E48:53"/><i><font color="Green">E116</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 <b>and </b><br/><a NAME="E49:53"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span> holds <br/><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E50:53"/><i><font color="Green">E118</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_13"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_13"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E4:53_13"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> 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="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:53_13"/><i><font color="Green">E121</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:53_13"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T26" target="_self">Th26</a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:53_13"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#T26" target="_self">Th26</a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E9:53_13"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:53_13"/><b>then </b>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> <a href="finseq_4.html#R2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="vectsp_6.html#T27" target="_self">Th27</a>, <a class="txt" href="vectsp_6.html#E8:53"><i><font color="Green">E70</font></i></a>, <a class="txt" href="vectsp_6.html#E10:53"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/>
<b>hence </b><a NAME="E11:53_13"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D3">FINSEQ_4:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E123</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:53_14"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_14_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<a NAME="E2:53_14_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.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>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#T26" target="_self">Th26</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:53_14_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E4:53_14_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D8" target="_self">Def8</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:53_14_1"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:53_14"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> )
 ;<br/><a NAME="E3:53_14"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.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>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:53_14"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="vectsp_6.html#T26" target="_self">Th26</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_15"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> 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="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E3:53_15"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D7" target="_self">Def7</a>, <a class="txt" href="vectsp_6.html#E8:53"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:53_15"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> <a href="finseq_4.html#R2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/><a NAME="E5:53_15"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="vectsp_6.html#E8:53"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_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">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E10:53"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_4.html#D3">FINSEQ_4:def 3</a></i>
;<br/><b>hence </b><a NAME="E7:53_15"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_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">x</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E53:53"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>
<a NAME="E54:53"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_16"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E2:53_16"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="vectsp_6.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>then </b><i><font color="Green">E129</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E8:53"><i><font color="Green">E70</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="vectsp_6.html#D8" target="_self">Def8</a>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E5:53_16"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E10:53"><i><font color="Green">E71</font></i></a>, <a class="txt" href="vectsp_6.html#E11:53"><i><font color="Green">E72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E55:53"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D8" target="_self">Def8</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E56:53"/><i><font color="Green">E131</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T26" target="_self">Th26</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E57:53"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E10:53"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_4.html#T75">FINSEQ_4:75</a></i>;<br/>
<a NAME="E58:53"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>, <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <b>by </b><i><a class="ref" href="vectsp_6.html#D8" target="_self">Def8</a>, <a class="txt" href="vectsp_6.html#E11:53"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <b>by </b><i><a class="txt" href="vectsp_6.html#E10:53"><i><font color="Green">E71</font></i></a>, <a class="txt" href="vectsp_6.html#E12:53"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/>
<a NAME="E61:53"/><i><font color="Green">E133</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E62:53"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">Fp</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <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">V</font> <b>by </b><i><a class="ref" href="finseq_2.html#T51">FINSEQ_2:51</a></i>;<br/>
<a NAME="E65:53"/><i><font color="Green">E134</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">Fp</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T9">RLVECT_2:9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E67:53"/><i><font color="Green">E135</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 <b>and </b><br/><a NAME="E68:53"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> holds <br/><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E69:53"/><i><font color="Green">E137</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_17"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_17"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E4:53_17"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="vectsp_6.html#E14:53"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:53_17"/><i><font color="Green">E140</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:53_17"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T35" target="_self">Th35</a>, <a class="txt" href="vectsp_6.html#E14:53"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:53_17"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#T35" target="_self">Th35</a>, <a class="txt" href="vectsp_6.html#E14:53"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E9:53_17"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:53_17"/><b>then </b>
( <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="finseq_4.html#R2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="vectsp_6.html#T37" target="_self">Th37</a>, <a class="txt" href="vectsp_6.html#E17:53"><i><font color="Green">E81</font></i></a>, <a class="ref" href="vectsp_6.html#D9" target="_self">Def9</a>, <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/>
<b>hence </b><a NAME="E11:53_17"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E15:53"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_4.html#D3">FINSEQ_4:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E142</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_18"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:53_18"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<a NAME="E2:53_18_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#T35" target="_self">Th35</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:53_18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E4:53_18_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:53_18_1"/>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:53_18"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> &amp; <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> )
 ;<br/><a NAME="E3:53_18"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:53_18"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="vectsp_6.html#T35" target="_self">Th35</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E143</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_19"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_19"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> 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="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E3:53_19"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D7" target="_self">Def7</a>, <a class="txt" href="vectsp_6.html#E17:53"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:53_19"/><b>then </b><i><font color="Green">E179</font></i>: 
<font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="finseq_4.html#R2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/><a NAME="E5:53_19"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="vectsp_6.html#E17:53"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="finseq_4.html#K1">&lt;-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T37" target="_self">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="vectsp_6.html#D9" target="_self">Def9</a>, <a class="ref" href="finseq_4.html#D3">FINSEQ_4:def 3</a></i>
;<br/><b>hence </b><a NAME="E7:53_19"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E72:53"/><b>then </b><i><font color="Green">E180</font></i>: 
<font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E14:53"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>
<a NAME="E73:53"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_20"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E2:53_20"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="vectsp_6.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>then </b><i><font color="Green">E182</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">R</font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E17:53"><i><font color="Green">E81</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">R</font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:53"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon">R</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E5:53_20"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9" target="_self">Def9</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E74:53"/><b>then </b><i><font color="Green">E183</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E75:53"/><i><font color="Green">E184</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T35" target="_self">Th35</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E76:53"/><b>then </b><i><font color="Green">E185</font></i>: 
<font color="Maroon">R</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D9" target="_self">Def9</a>, <a class="ref" href="finseq_4.html#T75">FINSEQ_4:75</a></i>;<br/>
<a NAME="E77:53"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>, <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <b>by </b><i><a class="ref" href="vectsp_6.html#D9" target="_self">Def9</a>, , <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/>
<a NAME="E80:53"/><i><font color="Green">E186</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E81:53"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">Hp</font> = <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">R</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="finseq_2.html#T51">FINSEQ_2:51</a></i>;<br/>
<a NAME="E84:53"/><i><font color="Green">E188</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">Hp</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T9">RLVECT_2:9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">Hp</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">I</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E86:53"/><i><font color="Green">E189</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 <b>and </b><br/><a NAME="E87:53"/><i><font color="Green">E190</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E88:53"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_21"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">I</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_21"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E4:53_21"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="vectsp_6.html#E19:53"><i><font color="Green">E89</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:53_21"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:53_21"/><b>then </b>
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">Hp</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="vectsp_6.html#E19:53"><i><font color="Green">E89</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:53_21"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E20:53"><i><font color="Green">E90</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">I</font> = <font color="Maroon">I</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E90:53"/><i><font color="Green">E193</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#T32" target="_self">Th32</a>, , <a class="ref" href="finseq_2.html#T48">FINSEQ_2:48</a></i>;<br/>
<a NAME="E91:53"/><b>then </b><i><font color="Green">E194</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E92:53"/><i><font color="Green">E195</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Hp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="finseq_2.html#T48">FINSEQ_2:48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_22"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_22"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font></span>)</span>
 ;<br/><a NAME="E2:53_22"/><b>then </b><i><font color="Green">E197</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Hp</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fp</font> )
 <b>by </b><i>, , <a class="txt" href="vectsp_6.html#E19:53"><i><font color="Green">E89</font></i></a>, <a class="txt" href="vectsp_6.html#E22:53"><i><font color="Green">E91</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="vectsp_6.html#E23:53"><i><font color="Green">E92</font></i></a></i>;<br/><b>set </b><font color="Maroon">v</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>;<br/><i><font color="Green">E198</font></i>: <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a>, </i>
;<br/><i><font color="Green">E199</font></i>: <font color="Maroon">Hp</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><a NAME="E6:53_22"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9" target="_self">Def9</a>, , <a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:53_22"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9" target="_self">Def9</a>, , <a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="vectsp_6.html#E27:53"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E15:53"><i><font color="Green">E80</font></i></a>, <a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E10:53_22"/><b>then </b><i><font color="Green">E203</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E28:53"><i><font color="Green">E97</font></i></a>, </i>;<br/><a NAME="E11:53_22"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">Fp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="funct_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">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E12:53_22"/><i><font color="Green">E205</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E13:53_22"/><b>then </b><i><font color="Green">E206</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E10:53"><i><font color="Green">E71</font></i></a>, <a class="txt" href="vectsp_6.html#E11:53"><i><font color="Green">E72</font></i></a>, <a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E14:53_22"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E10:53"><i><font color="Green">E71</font></i></a>, <a class="txt" href="vectsp_6.html#E11:53"><i><font color="Green">E72</font></i></a>, <a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a>, <a class="txt" href="vectsp_6.html#E33:53"><i><font color="Green">E100</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> 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="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E25:53"><i><font color="Green">E93</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/><b>hence </b><a NAME="E18:53_22"/>
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="txt" href="vectsp_6.html#E23:53"><i><font color="Green">E92</font></i></a>, , , <a class="txt" href="vectsp_6.html#E30:53"><i><font color="Green">E98</font></i></a>, <a class="txt" href="vectsp_6.html#E31:53"><i><font color="Green">E99</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E94:53"/><b>then </b><i><font color="Green">E208</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fp</font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E20:53"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E95:53"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E22:53"><i><font color="Green">E91</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E96:53"/>
 <a href="vectsp_6.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K5" target="_self">+</a> <font color="Maroon">L2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">L1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">L2</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="vectsp_6.html#D6" target="_self">Def6</a>, , <a class="ref" href="vectsp_6.html#T34" target="_self">Th34</a>, , , , <a class="txt" href="vectsp_6.html#E22:53"><i><font color="Green">E91</font></i></a>, <a class="txt" href="vectsp_6.html#E23:53"><i><font color="Green">E92</font></i></a>, <a class="ref" href="rlvect_2.html#T4">RLVECT_2:4</a></i>;<br/>


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