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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c3">L1</font>, <font color="Maroon" title="c4">L2</font> be    <a href="rmod_4.html#M1" title="RMOD_4:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>;<br/>







<b>consider </b><font color="Maroon" title="c5">F</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E2:69"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c5">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E3:69"/><i><font color="Green" title="E41">A2</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span>
 <b>and </b><br/><a NAME="E4:69"/><i><font color="Green" title="E42">A3</font></i>: 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span>
 <b>by </b><i><a class="ref" href="rmod_4.html#D9" target="_self" title="RMOD_4:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">G</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E6:69"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c6">G</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E7:69"/><i><font color="Green" title="E44">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font>
 <b>and </b><br/><a NAME="E8:69"/><i><font color="Green" title="E45">A6</font></i>: 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c3">L1</font>
 <b>by </b><i><a class="ref" href="rmod_4.html#D9" target="_self" title="RMOD_4:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">H</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E10:69"/><i><font color="Green" title="E46">A7</font></i>: 
<font color="Maroon" title="c7">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E11:69"/><i><font color="Green" title="E47">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font>
 <b>and </b><br/><a NAME="E12:69"/><i><font color="Green" title="E48">A9</font></i>: 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c4">L2</font>
 <b>by </b><i><a class="ref" href="rmod_4.html#D9" target="_self" title="RMOD_4:def.9">Def9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">A</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">C1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c10">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E14:69"/><i><font color="Green" title="E49">A10</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span>
 <b>and </b><br/><a NAME="E15:69"/><i><font color="Green" title="E50">A11</font></i>: 
<font color="Maroon" title="c10">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73" title="FINSEQ_4:th.73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">p</font> = <font color="Maroon" title="c10">p</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="rmod_4.html#E14:69"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E17:69"/><i><font color="Green" title="E51">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c11">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rmod_4.html#D8" target="_self" title="RMOD_4:def.8">Def8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:69_1"/><i><font color="Green" title="E52">A13</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">p</font>
 ;<br/><a NAME="E2:69_1"/><b>then </b>
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c11">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E17:69"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:69_1"/><b>then </b><i><font color="Green" title="E53">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rmod_4.html#D8" target="_self" title="RMOD_4:def.8">Def8</a></i>;<br/><a NAME="E4:69_1"/>
<font color="Maroon" title="c11">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:69_1"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:69_1"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E14:69"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="rmod_4.html#E1:69_1"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:69_1"/><b>then </b>
not <font color="Maroon" title="c11">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:69_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E3:69_1"><i><font color="Green" title="E53">A14</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E52">A15</font></i>:  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c11">p</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E17:69"><i><font color="Green" title="E51">A12</font></i></a>, <a class="txt" href="rmod_4.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font>
<b>by </b><i><a class="ref" href="mod_1.html#T37" title="MOD_1:th.37">MOD_1:37</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c12">GG</font> = <font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font>;<br/>
<b>set </b><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>;<br/>
<i><font color="Green" title="E53">A16</font></i>:  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> = 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#T37" target="_self" title="RMOD_4:th.37">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c6">G</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E19:69"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c6">G</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T7" title="VECTSP_1:th.7">VECTSP_1:7</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c14">C2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c15">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E22:69"/><i><font color="Green" title="E54">A17</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span>
 <b>and </b><br/><a NAME="E23:69"/><i><font color="Green" title="E55">A18</font></i>: 
<font color="Maroon" title="c15">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73" title="FINSEQ_4:th.73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">q</font> = <font color="Maroon" title="c15">q</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="rmod_4.html#E22:69"><i><font color="Green" title="E54">A17</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E25:69"/><i><font color="Green" title="E56">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c16">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rmod_4.html#D8" target="_self" title="RMOD_4:def.8">Def8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:69_4"/><i><font color="Green" title="E57">A20</font></i>: 
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">q</font>
 ;<br/><a NAME="E2:69_4"/><b>then </b>
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c16">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E25:69"><i><font color="Green" title="E56">A19</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:69_4"/><b>then </b><i><font color="Green" title="E58">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rmod_4.html#D8" target="_self" title="RMOD_4:def.8">Def8</a></i>;<br/><a NAME="E4:69_4"/>
<font color="Maroon" title="c16">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:69_4"><i><font color="Green" title="E57">A20</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:69_4"/><b>then </b>
<font color="Maroon" title="c16">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E22:69"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="rmod_4.html#E1:69_4"><i><font color="Green" title="E57">A20</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:69_4"/><b>then </b>
not <font color="Maroon" title="c16">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:69_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E3:69_4"><i><font color="Green" title="E58">A21</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E57">A22</font></i>:  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c16">q</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E25:69"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="rmod_4.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font>
<b>by </b><i><a class="ref" href="mod_1.html#T37" title="MOD_1:th.37">MOD_1:37</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c17">HH</font> = <font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font>;<br/>
<b>set </b><font color="Maroon" title="c18">h</font> = <font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>;<br/>
<i><font color="Green" title="E58">A23</font></i>:  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> = 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#T37" target="_self" title="RMOD_4:th.37">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E27:69"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c7">H</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T7" title="VECTSP_1:th.7">VECTSP_1:7</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c19">C3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c20">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E30:69"/><i><font color="Green" title="E59">A24</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E31:69"/><i><font color="Green" title="E60">A25</font></i>: 
<font color="Maroon" title="c20">r</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73" title="FINSEQ_4:th.73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">r</font> = <font color="Maroon" title="c20">r</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="rmod_4.html#E30:69"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E33:69"/><i><font color="Green" title="E61">A26</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c21">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rmod_4.html#D8" target="_self" title="RMOD_4:def.8">Def8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:69_7"/><i><font color="Green" title="E62">A27</font></i>: 
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c21">r</font>
 ;<br/><a NAME="E2:69_7"/><b>then </b>
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c21">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E33:69"><i><font color="Green" title="E61">A26</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:69_7"/><b>then </b><i><font color="Green" title="E63">A28</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rmod_4.html#D8" target="_self" title="RMOD_4:def.8">Def8</a></i>;<br/><a NAME="E4:69_7"/>
<font color="Maroon" title="c21">r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:69_7"><i><font color="Green" title="E62">A27</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:69_7"/><b>then </b>
<font color="Maroon" title="c21">r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E30:69"><i><font color="Green" title="E59">A24</font></i></a>, <a class="txt" href="rmod_4.html#E1:69_7"><i><font color="Green" title="E62">A27</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:69_7"/><b>then </b>
not <font color="Maroon" title="c21">r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:69_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E3:69_7"><i><font color="Green" title="E63">A28</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E62">A29</font></i>:  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c21">r</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E33:69"><i><font color="Green" title="E61">A26</font></i></a>, <a class="txt" href="rmod_4.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font>
<b>by </b><i><a class="ref" href="mod_1.html#T37" title="MOD_1:th.37">MOD_1:37</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c22">FF</font> = <font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font>;<br/>
<b>set </b><font color="Maroon" title="c23">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>;<br/>
<i><font color="Green" title="E63">A30</font></i>:  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span> = 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#T37" target="_self" title="RMOD_4:th.37">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E35:69"><i><font color="Green" title="E62">A29</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <font color="Maroon" title="c5">F</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T7" title="VECTSP_1:th.7">VECTSP_1:7</a></i>
;<br/>
<a NAME="E37:69"/><i><font color="Green" title="E64">A31</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_10"/>
not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">p</font>
 ;<br/>

<a NAME="E2:69_10"/><b>then </b><i><font color="Green" title="E65">A32</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">p</font></span>)</span>;<br/>
<a NAME="E4:69_10"/>
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font> &amp; <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E7:69"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="rmod_4.html#E14:69"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="rmod_4.html#E2:69_10"><i><font color="Green" title="E65">A32</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:69_10"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:69"/><i><font color="Green" title="E65">A33</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">H</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_11"/>
not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">H</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q</font>
 ;<br/>

<a NAME="E2:69_11"/><b>then </b><i><font color="Green" title="E66">A34</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q</font></span>)</span>;<br/>
<a NAME="E4:69_11"/>
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font> &amp; <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E11:69"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="rmod_4.html#E22:69"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="rmod_4.html#E2:69_11"><i><font color="Green" title="E66">A34</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:69_11"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:69"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_12"/>
not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">r</font>
 ;<br/>

<a NAME="E2:69_12"/><b>then </b><i><font color="Green" title="E66">A35</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">r</font></span>)</span>;<br/>
<a NAME="E4:69_12"/>
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> &amp; <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E3:69"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="rmod_4.html#E30:69"><i><font color="Green" title="E59">A24</font></i></a>, <a class="txt" href="rmod_4.html#E2:69_12"><i><font color="Green" title="E66">A35</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:69_12"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E40:69"/><b>then </b><i><font color="Green" title="E66">A36</font></i>: 
( <font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E2:69"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="rmod_4.html#E6:69"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="rmod_4.html#E10:69"><i><font color="Green" title="E46">A7</font></i></a>, <a class="txt" href="rmod_4.html#E15:69"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="rmod_4.html#E23:69"><i><font color="Green" title="E55">A18</font></i></a>, <a class="txt" href="rmod_4.html#E31:69"><i><font color="Green" title="E60">A25</font></i></a>, <a class="txt" href="rmod_4.html#E37:69"><i><font color="Green" title="E64">A31</font></i></a>, <a class="txt" href="rmod_4.html#E38:69"><i><font color="Green" title="E65">A33</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E41:69"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">p</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">q</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E42:69"/><b>then </b><i><font color="Green" title="E67">A37</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E3:69"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="rmod_4.html#E7:69"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="rmod_4.html#E11:69"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="rmod_4.html#E14:69"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="rmod_4.html#E22:69"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="rmod_4.html#E30:69"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>;<br/>
<a NAME="E43:69"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E44:69"/><b>then </b>
(  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span> &amp;  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span> &amp;  <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E45:69"/><b>then </b><i><font color="Green" title="E68">A38</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E42:69"><i><font color="Green" title="E67">A37</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E46:69"/><b>then </b><i><font color="Green" title="E69">A39</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E40:69"><i><font color="Green" title="E66">A36</font></i></a>, <a class="ref" href="finseq_1.html#T65" title="FINSEQ_1:th.65">FINSEQ_1:65</a></i>;<br/>
<a NAME="E47:69"/><b>then </b><i><font color="Green" title="E70">A40</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="finseq_4.html#K1" title="FINSEQ_4:func.1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c24">P</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E49:69"/><i><font color="Green" title="E71">A41</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 <b>and </b><br/><a NAME="E50:69"/><i><font color="Green" title="E72">A42</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E51:69"/><i><font color="Green" title="E73">A43</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_13"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c26">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:69_13"/><i><font color="Green" title="E74">A44</font></i>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">P</font>
 <b>and </b><br/><a NAME="E4:69_13"/><i><font color="Green" title="E75">A45</font></i>: 
<font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">y</font> = <font color="Maroon" title="c26">y</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="rmod_4.html#E3:69_13"><i><font color="Green" title="E74">A44</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:69_13"/><i><font color="Green" title="E76">A46</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:69_13"/><i><font color="Green" title="E77">A47</font></i>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E49:69"><i><font color="Green" title="E71">A41</font></i></a>, <a class="txt" href="rmod_4.html#E3:69_13"><i><font color="Green" title="E74">A44</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:69_13"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E45:69"><i><font color="Green" title="E68">A38</font></i></a>, <a class="txt" href="rmod_4.html#E47:69"><i><font color="Green" title="E70">A40</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:69_13"/><b>then </b>
( <font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="finseq_4.html#K1" title="FINSEQ_4:func.1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font> <a href="finseq_4.html#R2" title="FINSEQ_4:pred.2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E40:69"><i><font color="Green" title="E66">A36</font></i></a>, <a class="txt" href="rmod_4.html#E50:69"><i><font color="Green" title="E72">A42</font></i></a>, <a class="txt" href="rmod_4.html#E6:69_13"><i><font color="Green" title="E76">A46</font></i></a>, <a class="txt" href="rmod_4.html#E7:69_13"><i><font color="Green" title="E77">A47</font></i></a>, <a class="ref" href="finseq_4.html#T10" title="FINSEQ_4:th.10">FINSEQ_4:10</a></i>;<br/>
<b>hence </b><a NAME="E10:69_13"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E4:69_13"><i><font color="Green" title="E75">A45</font></i></a>, <a class="ref" href="finseq_4.html#D3" title="FINSEQ_4:def.3">FINSEQ_4:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E74">A48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:69_14"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> implies ( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">P</font> &amp; <font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_14_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E2:69_14_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">P</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E46:69"><i><font color="Green" title="E69">A39</font></i></a>, <a class="txt" href="rmod_4.html#E49:69"><i><font color="Green" title="E71">A41</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>

<a NAME="E3:69_14_1"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> &amp; <font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">P</font> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E51:69"><i><font color="Green" title="E73">A43</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:69_14_1"/>
<font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:69_14"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">P</font> &amp; <font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> )
 ;<br/><a NAME="E3:69_14"/><b>then </b>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:69_14"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E46:69"><i><font color="Green" title="E69">A39</font></i></a>, <a class="txt" href="rmod_4.html#E49:69"><i><font color="Green" title="E71">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E75">A49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:69_15"/><i><font color="Green" title="E76">A50</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c26">n</font> = <font color="Maroon" title="c25">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><a NAME="E3:69_15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E45:69"><i><font color="Green" title="E68">A38</font></i></a>, <a class="txt" href="rmod_4.html#E1:69_15"><i><font color="Green" title="E76">A50</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:69_15"/><b>then </b><i><font color="Green" title="E77">A51</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font> <a href="finseq_4.html#R2" title="FINSEQ_4:pred.2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">n</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E40:69"><i><font color="Green" title="E66">A36</font></i></a>, <a class="ref" href="finseq_4.html#T10" title="FINSEQ_4:th.10">FINSEQ_4:10</a></i>;<br/><a NAME="E5:69_15"/>
<font color="Maroon" title="c26">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E46:69"><i><font color="Green" title="E69">A39</font></i></a>, <a class="txt" href="rmod_4.html#E1:69_15"><i><font color="Green" title="E76">A50</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="finseq_4.html#K1" title="FINSEQ_4:func.1">&lt;-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E50:69"><i><font color="Green" title="E72">A42</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">n</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E4:69_15"><i><font color="Green" title="E77">A51</font></i></a>, <a class="ref" href="finseq_4.html#D3" title="FINSEQ_4:def.3">FINSEQ_4:def 3</a></i>
;<br/><b>hence </b><a NAME="E7:69_15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E54:69"/><b>then </b><i><font color="Green" title="E76">A52</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c24">P</font>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E52:69"><i><font color="Green" title="E74">A48</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>
<a NAME="E55:69"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_16"/><i><font color="Green" title="E77">A53</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c26">f</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>;<br/>
<a NAME="E2:69_16"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E40:69"><i><font color="Green" title="E66">A36</font></i></a>, <a class="txt" href="rmod_4.html#E45:69"><i><font color="Green" title="E68">A38</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
<b>then </b><i><font color="Green" title="E78">A54</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E40:69"><i><font color="Green" title="E66">A36</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">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" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c24">P</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E54:69"><i><font color="Green" title="E76">A52</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c24">P</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E40:69"><i><font color="Green" title="E66">A36</font></i></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c24">P</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E51:69"><i><font color="Green" title="E73">A43</font></i></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E5:69_16"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">P</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:69_16"><i><font color="Green" title="E77">A53</font></i></a>, <a class="txt" href="rmod_4.html#E3:69_16"><i><font color="Green" title="E78">A54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E56:69"/><b>then </b><i><font color="Green" title="E77">A55</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">P</font>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E51:69"><i><font color="Green" title="E73">A43</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E57:69"/><i><font color="Green" title="E78">A56</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E49:69"><i><font color="Green" title="E71">A41</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E58:69"/><b>then </b><i><font color="Green" title="E79">A57</font></i>: 
<font color="Maroon" title="c24">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E56:69"><i><font color="Green" title="E77">A55</font></i></a>, <a class="ref" href="finseq_4.html#T75" title="FINSEQ_4:th.75">FINSEQ_4:75</a></i>;<br/>
<a NAME="E59:69"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c25">P</font> = <font color="Maroon" title="c24">P</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>, <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <b>by </b><i><a class="txt" href="rmod_4.html#E51:69"><i><font color="Green" title="E73">A43</font></i></a>, <a class="txt" href="rmod_4.html#E57:69"><i><font color="Green" title="E78">A56</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E61:69"/><i><font color="Green" title="E80">A58</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rmod_4.html#D8" target="_self" title="RMOD_4:def.8">Def8</a></i>;<br/>
<a NAME="E62:69"/><b>then </b><i><font color="Green" title="E81">A59</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">P</font> = <font color="Maroon" title="c25">P</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span>, <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c27">Fp</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c26">P</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="finseq_2.html#T51" title="FINSEQ_2:th.51">FINSEQ_2:51</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">P</font> = <font color="Maroon" title="c26">P</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="rmod_4.html#E56:69"><i><font color="Green" title="E77">A55</font></i></a>, <a class="txt" href="rmod_4.html#E58:69"><i><font color="Green" title="E79">A57</font></i></a>, <a class="txt" href="rmod_4.html#E62:69"><i><font color="Green" title="E81">A59</font></i></a>, <a class="ref" href="funct_2.html#T83" title="FUNCT_2:th.83">FUNCT_2:83</a></i>;<br/>
<a NAME="E66:69"/><i><font color="Green" title="E82">A60</font></i>: 
<font color="Maroon" title="c27">Fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c28">P</font>
 
;<br/>
<a NAME="E67:69"/><b>then </b><i><font color="Green" title="E83">A61</font></i>: 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c27">Fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T9" title="RLVECT_2:th.9">RLVECT_2:9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="finseq_4.html#K1" title="FINSEQ_4:func.1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c29">R</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E69:69"/><i><font color="Green" title="E84">A62</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 <b>and </b><br/><a NAME="E70:69"/><i><font color="Green" title="E85">A63</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c29">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E71:69"/><i><font color="Green" title="E86">A64</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_17"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c31">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:69_17"/><i><font color="Green" title="E87">A65</font></i>: 
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c29">R</font>
 <b>and </b><br/><a NAME="E4:69_17"/><i><font color="Green" title="E88">A66</font></i>: 
<font color="Maroon" title="c29">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">y</font> = <font color="Maroon" title="c31">y</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="rmod_4.html#E3:69_17"><i><font color="Green" title="E87">A65</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:69_17"/><i><font color="Green" title="E89">A67</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:69_17"/><i><font color="Green" title="E90">A68</font></i>: 
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E69:69"><i><font color="Green" title="E84">A62</font></i></a>, <a class="txt" href="rmod_4.html#E3:69_17"><i><font color="Green" title="E87">A65</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:69_17"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E45:69"><i><font color="Green" title="E68">A38</font></i></a>, <a class="txt" href="rmod_4.html#E47:69"><i><font color="Green" title="E70">A40</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:69_17"/><b>then </b>
( <font color="Maroon" title="c29">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="finseq_4.html#K1" title="FINSEQ_4:func.1">&lt;-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">y</font></span>)</span> &amp; <font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font> <a href="finseq_4.html#R2" title="FINSEQ_4:pred.2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">y</font> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E40:69"><i><font color="Green" title="E66">A36</font></i></a>, <a class="txt" href="rmod_4.html#E70:69"><i><font color="Green" title="E85">A63</font></i></a>, <a class="txt" href="rmod_4.html#E6:69_17"><i><font color="Green" title="E89">A67</font></i></a>, <a class="txt" href="rmod_4.html#E7:69_17"><i><font color="Green" title="E90">A68</font></i></a>, <a class="ref" href="finseq_4.html#T10" title="FINSEQ_4:th.10">FINSEQ_4:10</a></i>;<br/>
<b>hence </b><a NAME="E10:69_17"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E4:69_17"><i><font color="Green" title="E88">A66</font></i></a>, <a class="ref" href="finseq_4.html#D3" title="FINSEQ_4:def.3">FINSEQ_4:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E87">A69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_18"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:69_18"/>
( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> implies ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c29">R</font> &amp; <font color="Maroon" title="c29">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_18_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>
 ;<br/>

<a NAME="E2:69_18_1"/><b>then </b>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E46:69"><i><font color="Green" title="E69">A39</font></i></a>, <a class="txt" href="rmod_4.html#E69:69"><i><font color="Green" title="E84">A62</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:69_18_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c29">R</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E4:69_18_1"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> &amp; <font color="Maroon" title="c29">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">R</font> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E71:69"><i><font color="Green" title="E86">A64</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:69_18_1"/>
<font color="Maroon" title="c29">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:69_18"/>
( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c29">R</font> &amp; <font color="Maroon" title="c29">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> )
 ;<br/><a NAME="E3:69_18"/><b>then </b>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:69_18"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E46:69"><i><font color="Green" title="E69">A39</font></i></a>, <a class="txt" href="rmod_4.html#E69:69"><i><font color="Green" title="E84">A62</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E88">A70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_19"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:69_19"/><i><font color="Green" title="E89">A71</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c31">n</font> = <font color="Maroon" title="c30">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><a NAME="E3:69_19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E45:69"><i><font color="Green" title="E68">A38</font></i></a>, <a class="txt" href="rmod_4.html#E1:69_19"><i><font color="Green" title="E89">A71</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:69_19"/><b>then </b><i><font color="Green" title="E90">A72</font></i>: 
<font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font> <a href="finseq_4.html#R2" title="FINSEQ_4:pred.2">just_once_values</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">n</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E40:69"><i><font color="Green" title="E66">A36</font></i></a>, <a class="ref" href="finseq_4.html#T10" title="FINSEQ_4:th.10">FINSEQ_4:10</a></i>;<br/><a NAME="E5:69_19"/>
<font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E46:69"><i><font color="Green" title="E69">A39</font></i></a>, <a class="txt" href="rmod_4.html#E1:69_19"><i><font color="Green" title="E89">A71</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="finseq_4.html#K1" title="FINSEQ_4:func.1">&lt;-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E70:69"><i><font color="Green" title="E85">A63</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">n</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E4:69_19"><i><font color="Green" title="E90">A72</font></i></a>, <a class="ref" href="finseq_4.html#D3" title="FINSEQ_4:def.3">FINSEQ_4:def 3</a></i>
;<br/><b>hence </b><a NAME="E7:69_19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E74:69"/><b>then </b><i><font color="Green" title="E89">A73</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c29">R</font>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E72:69"><i><font color="Green" title="E87">A69</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>
<a NAME="E75:69"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_20"/><i><font color="Green" title="E90">A74</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c31">f</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>;<br/>
<a NAME="E2:69_20"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E40:69"><i><font color="Green" title="E66">A36</font></i></a>, <a class="txt" href="rmod_4.html#E45:69"><i><font color="Green" title="E68">A38</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
<b>then </b><i><font color="Green" title="E91">A75</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E40:69"><i><font color="Green" title="E66">A36</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">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" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c29">R</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E74:69"><i><font color="Green" title="E89">A73</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c29">R</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E40:69"><i><font color="Green" title="E66">A36</font></i></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c29">R</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E71:69"><i><font color="Green" title="E86">A64</font></i></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E5:69_20"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">R</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:69_20"><i><font color="Green" title="E90">A74</font></i></a>, <a class="txt" href="rmod_4.html#E3:69_20"><i><font color="Green" title="E91">A75</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E76:69"/><b>then </b><i><font color="Green" title="E90">A76</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">R</font>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E71:69"><i><font color="Green" title="E86">A64</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E77:69"/><i><font color="Green" title="E91">A77</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c29">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E69:69"><i><font color="Green" title="E84">A62</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E78:69"/><b>then </b><i><font color="Green" title="E92">A78</font></i>: 
<font color="Maroon" title="c29">R</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E76:69"><i><font color="Green" title="E90">A76</font></i></a>, <a class="ref" href="finseq_4.html#T75" title="FINSEQ_4:th.75">FINSEQ_4:75</a></i>;<br/>
<a NAME="E79:69"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">R</font> = <font color="Maroon" title="c29">R</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>, <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <b>by </b><i><a class="txt" href="rmod_4.html#E71:69"><i><font color="Green" title="E86">A64</font></i></a>, <a class="txt" href="rmod_4.html#E77:69"><i><font color="Green" title="E91">A77</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E81:69"/><i><font color="Green" title="E93">A79</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rmod_4.html#D8" target="_self" title="RMOD_4:def.8">Def8</a></i>;<br/>
<a NAME="E82:69"/><b>then </b><i><font color="Green" title="E94">A80</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c31">R</font> = <font color="Maroon" title="c30">R</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span>, <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c32">Hp</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c31">R</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="finseq_2.html#T51" title="FINSEQ_2:th.51">FINSEQ_2:51</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">R</font> = <font color="Maroon" title="c31">R</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="rmod_4.html#E76:69"><i><font color="Green" title="E90">A76</font></i></a>, <a class="txt" href="rmod_4.html#E78:69"><i><font color="Green" title="E92">A78</font></i></a>, <a class="txt" href="rmod_4.html#E82:69"><i><font color="Green" title="E94">A80</font></i></a>, <a class="ref" href="funct_2.html#T83" title="FUNCT_2:th.83">FUNCT_2:83</a></i>;<br/>
<a NAME="E86:69"/><i><font color="Green" title="E95">A81</font></i>: 
<font color="Maroon" title="c32">Hp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c33">R</font>
 
;<br/>
<a NAME="E87:69"/><b>then </b><i><font color="Green" title="E96">A82</font></i>: 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c32">Hp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T9" title="RLVECT_2:th.9">RLVECT_2:9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1</span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">Hp</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c34">I</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E89:69"/><i><font color="Green" title="E97">A83</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span>
 <b>and </b><br/><a NAME="E90:69"/><i><font color="Green" title="E98">A84</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c34">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E91:69"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c35">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_21"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">I</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c36">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:69_21"/><i><font color="Green" title="E99">A85</font></i>: 
<font color="Maroon" title="c36">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c34">I</font>
 <b>and </b><br/><a NAME="E4:69_21"/><i><font color="Green" title="E100">A86</font></i>: 
<font color="Maroon" title="c34">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c37">y</font> = <font color="Maroon" title="c36">y</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="rmod_4.html#E3:69_21"><i><font color="Green" title="E99">A85</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:69_21"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c34">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:69_21"/><b>then </b>
<font color="Maroon" title="c34">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">y</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">Hp</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E89:69"><i><font color="Green" title="E97">A83</font></i></a>, <a class="txt" href="rmod_4.html#E90:69"><i><font color="Green" title="E98">A84</font></i></a>, <a class="txt" href="rmod_4.html#E3:69_21"><i><font color="Green" title="E99">A85</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:69_21"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E4:69_21"><i><font color="Green" title="E100">A86</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c35">I</font> = <font color="Maroon" title="c34">I</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E93:69"/><i><font color="Green" title="E99">A87</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">Fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">I</font>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E46:69"><i><font color="Green" title="E69">A39</font></i></a>, <a class="txt" href="rmod_4.html#E61:69"><i><font color="Green" title="E80">A58</font></i></a>, <a class="txt" href="rmod_4.html#E66:69"><i><font color="Green" title="E82">A60</font></i></a>, <a class="txt" href="rmod_4.html#E89:69"><i><font color="Green" title="E97">A83</font></i></a>, <a class="ref" href="finseq_2.html#T48" title="FINSEQ_2:th.48">FINSEQ_2:48</a></i>;<br/>
<a NAME="E94:69"/><b>then </b><i><font color="Green" title="E100">A88</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c35">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">I</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c27">Fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">I</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E95:69"/><i><font color="Green" title="E101">A89</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">Hp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E46:69"><i><font color="Green" title="E69">A39</font></i></a>, <a class="txt" href="rmod_4.html#E81:69"><i><font color="Green" title="E93">A79</font></i></a>, <a class="txt" href="rmod_4.html#E86:69"><i><font color="Green" title="E95">A81</font></i></a>, <a class="ref" href="rmod_4.html#D8" target="_self" title="RMOD_4:def.8">Def8</a>, <a class="ref" href="finseq_2.html#T48" title="FINSEQ_2:th.48">FINSEQ_2:48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_22"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c36">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:69_22"/><i><font color="Green" title="E102">A90</font></i>: 
<font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">I</font></span>)</span>
 ;<br/><a NAME="E2:69_22"/><b>then </b><i><font color="Green" title="E103">A91</font></i>: 
( <font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c32">Hp</font> &amp; <font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> &amp; <font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> &amp; <font color="Maroon" title="c36">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c27">Fp</font> )
 <b>by </b><i><a class="txt" href="rmod_4.html#E46:69"><i><font color="Green" title="E69">A39</font></i></a>, <a class="txt" href="rmod_4.html#E89:69"><i><font color="Green" title="E97">A83</font></i></a>, <a class="txt" href="rmod_4.html#E93:69"><i><font color="Green" title="E99">A87</font></i></a>, <a class="txt" href="rmod_4.html#E95:69"><i><font color="Green" title="E101">A89</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c37">k</font> = <font color="Maroon" title="c36">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="rmod_4.html#E1:69_22"><i><font color="Green" title="E102">A90</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c38">v</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font>;<br/><i><font color="Green" title="E104">A92</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a>, <a class="ref" href="rmod_4.html#D8" target="_self" title="RMOD_4:def.8">Def8</a></i>
;<br/><i><font color="Green" title="E105">A93</font></i>: <font color="Maroon" title="c32">Hp</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c33">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E6:69_22"/><i><font color="Green" title="E106">A94</font></i>: 
<font color="Maroon" title="c33">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c33">R</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E76:69"><i><font color="Green" title="E90">A76</font></i></a>, <a class="txt" href="rmod_4.html#E77:69"><i><font color="Green" title="E91">A77</font></i></a>, <a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:69_22"/><i><font color="Green" title="E107">A95</font></i>: 
<font color="Maroon" title="c33">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E76:69"><i><font color="Green" title="E90">A76</font></i></a>, <a class="txt" href="rmod_4.html#E77:69"><i><font color="Green" title="E91">A77</font></i></a>, <a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c39">j</font> = <font color="Maroon" title="c33">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="rmod_4.html#E6:69_22"><i><font color="Green" title="E106">A94</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E73:69"><i><font color="Green" title="E88">A70</font></i></a>, <a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E10:69_22"/><b>then </b><i><font color="Green" title="E108">A96</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E7:69_22"><i><font color="Green" title="E107">A95</font></i></a>, <a class="ref" href="rmod_4.html#T32" target="_self" title="RMOD_4:th.32">Th32</a></i>;<br/><a NAME="E11:69_22"/><i><font color="Green" title="E109">A97</font></i>: 
<font color="Maroon" title="c28">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c28">P</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E47:69"><i><font color="Green" title="E70">A40</font></i></a>, <a class="txt" href="rmod_4.html#E56:69"><i><font color="Green" title="E77">A55</font></i></a>, <a class="txt" href="rmod_4.html#E57:69"><i><font color="Green" title="E78">A56</font></i></a>, <a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:69_22"/><i><font color="Green" title="E110">A98</font></i>: 
<font color="Maroon" title="c28">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E47:69"><i><font color="Green" title="E70">A40</font></i></a>, <a class="txt" href="rmod_4.html#E56:69"><i><font color="Green" title="E77">A55</font></i></a>, <a class="txt" href="rmod_4.html#E57:69"><i><font color="Green" title="E78">A56</font></i></a>, <a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c40">l</font> = <font color="Maroon" title="c28">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="rmod_4.html#E11:69_22"><i><font color="Green" title="E109">A97</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E53:69"><i><font color="Green" title="E75">A49</font></i></a>, <a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>then </b><i><font color="Green" title="E111">A99</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">l</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E12:69_22"><i><font color="Green" title="E110">A98</font></i></a>, <a class="ref" href="rmod_4.html#T32" target="_self" title="RMOD_4:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#D11" target="_self" title="RMOD_4:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_2.html#D23" title="VECTSP_2:def.23">VECTSP_2:def 23</a></i>
;<br/><b>thus </b><font color="Maroon" title="c35">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E89:69"><i><font color="Green" title="E97">A83</font></i></a>, <a class="txt" href="rmod_4.html#E90:69"><i><font color="Green" title="E98">A84</font></i></a>, <a class="txt" href="rmod_4.html#E1:69_22"><i><font color="Green" title="E102">A90</font></i></a>, <a class="txt" href="rmod_4.html#E4:69_22"><i><font color="Green" title="E104">A92</font></i></a>, <a class="txt" href="rmod_4.html#E5:69_22"><i><font color="Green" title="E105">A93</font></i></a>, <a class="txt" href="rmod_4.html#E10:69_22"><i><font color="Green" title="E108">A96</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c27">Fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">x</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E2:69_22"><i><font color="Green" title="E103">A91</font></i></a>, <a class="txt" href="rmod_4.html#E15:69_22"><i><font color="Green" title="E111">A99</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E97:69"/><b>then </b><i><font color="Green" title="E102">A100</font></i>: 
<font color="Maroon" title="c35">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">Fp</font>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E94:69"><i><font color="Green" title="E100">A88</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E98:69"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K4" title="RMOD_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E95:69"><i><font color="Green" title="E101">A89</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E99:69"/>
 <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c4">L2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c3">L1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Maroon" title="c4">L2</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E4:69"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="rmod_4.html#E8:69"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="rmod_4.html#E12:69"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="rmod_4.html#E20:69"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="rmod_4.html#E28:69"><i><font color="Green" title="E58">A23</font></i></a>, <a class="txt" href="rmod_4.html#E36:69"><i><font color="Green" title="E63">A30</font></i></a>, <a class="txt" href="rmod_4.html#E67:69"><i><font color="Green" title="E83">A61</font></i></a>, <a class="txt" href="rmod_4.html#E87:69"><i><font color="Green" title="E96">A82</font></i></a>, <a class="txt" href="rmod_4.html#E89:69"><i><font color="Green" title="E97">A83</font></i></a>, <a class="txt" href="rmod_4.html#E90:69"><i><font color="Green" title="E98">A84</font></i></a>, <a class="txt" href="rmod_4.html#E95:69"><i><font color="Green" title="E101">A89</font></i></a>, <a class="txt" href="rmod_4.html#E97:69"><i><font color="Green" title="E102">A100</font></i></a>, <a class="ref" href="rlvect_2.html#T4" title="RLVECT_2:th.4">RLVECT_2:4</a></i>;<br/>


</div>
