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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">L1</font>, <font color="Olive" title="b3">L2</font> being   <a href="rmod_4.html#M1" title="RMOD_4:mode.1">Linear_Combination</a> of <font color="Olive" title="b1">V</font> holds   <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Olive" title="b3">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="Olive" title="b2">L1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b3">L2</font></span>)</span></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L1</font>, <font color="Olive" title="b2">L2</font> being   <a href="rmod_4.html#M1" title="RMOD_4:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> holds   <a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L1</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Olive" title="b2">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="Olive" title="b1">L1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K5" title="RMOD_4:func.5">Sum</a> <font color="Olive" title="b2">L2</font></span>)</span></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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#K3" title="RLVECT_1:func.3">+</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></span><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="E2:68"/><i><font color="Green" title="E44">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="E3:68"/><i><font color="Green" title="E45">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="E4:68"/><i><font color="Green" title="E46">A3</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">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="E6:68"/><i><font color="Green" title="E47">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="E7:68"/><i><font color="Green" title="E48">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="E8:68"/><i><font color="Green" title="E49">A6</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">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="E10:68"/><i><font color="Green" title="E50">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="E11:68"/><i><font color="Green" title="E51">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="E12:68"/><i><font color="Green" title="E52">A9</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">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="E14:68"/><i><font color="Green" title="E53">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="E15:68"/><i><font color="Green" title="E54">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="#E14:68"><i><font color="Green" title="E53">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="E17:68"/><i><font color="Green" title="E55">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="68_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">p</font> implies <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:68_1"/><i><font color="Green" title="E56">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><br/><a NAME="E2:68_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#NK2" title="RELAT_1:NK.2">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="#E17:68"><i><font color="Green" title="E55">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="E3:68_1"/><b>then </b><i><font color="Green" title="E57">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="E4:68_1"/>
<font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="#E1:68_1"><i><font color="Green" title="E56">A13</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E5:68_1"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="#E14:68"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="#E1:68_1"><i><font color="Green" title="E56">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="E6:68_1"/><b>then </b>
not <font color="Maroon" title="c11">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E7:68_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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:68_1"><i><font color="Green" title="E57">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E56">A15</font></i>:  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">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#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="#E17:68"><i><font color="Green" title="E55">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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
<b>by </b><i><a class="ref" href="vectsp_2.html#T90" title="VECTSP_2:th.90">VECTSP_2:90</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="E57">A16</font></i>:  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">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#K7" title="RLVECT_1:func.7">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#K7" title="RLVECT_1:func.7">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#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="txt" href="#E19:68"><i><font color="Green" title="E56">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#K7" title="RLVECT_1:func.7">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="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 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="E22:68"/><i><font color="Green" title="E58">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="E23:68"/><i><font color="Green" title="E59">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="#E22:68"><i><font color="Green" title="E58">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="E25:68"/><i><font color="Green" title="E60">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="68_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">q</font> implies <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:68_4"/><i><font color="Green" title="E61">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><br/><a NAME="E2:68_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#NK2" title="RELAT_1:NK.2">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="#E25:68"><i><font color="Green" title="E60">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="E3:68_4"/><b>then </b><i><font color="Green" title="E62">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="E4:68_4"/>
<font color="Maroon" title="c16">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="#E1:68_4"><i><font color="Green" title="E61">A20</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E5:68_4"/><b>then </b>
<font color="Maroon" title="c16">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="#E22:68"><i><font color="Green" title="E58">A17</font></i></a>, <a class="txt" href="#E1:68_4"><i><font color="Green" title="E61">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="E6:68_4"/><b>then </b>
not <font color="Maroon" title="c16">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E7:68_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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:68_4"><i><font color="Green" title="E62">A21</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E61">A22</font></i>:  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">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#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c16">q</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="#E25:68"><i><font color="Green" title="E60">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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
<b>by </b><i><a class="ref" href="vectsp_2.html#T90" title="VECTSP_2:th.90">VECTSP_2:90</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="E62">A23</font></i>:  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">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#K7" title="RLVECT_1:func.7">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#K7" title="RLVECT_1:func.7">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#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="txt" href="#E27:68"><i><font color="Green" title="E61">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#K7" title="RLVECT_1:func.7">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="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 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="E30:68"/><i><font color="Green" title="E63">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="E31:68"/><i><font color="Green" title="E64">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="#E30:68"><i><font color="Green" title="E63">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="E33:68"/><i><font color="Green" title="E65">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="68_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">r</font> implies <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:68_7"/><i><font color="Green" title="E66">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><br/><a NAME="E2:68_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#NK2" title="RELAT_1:NK.2">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="#E33:68"><i><font color="Green" title="E65">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="E3:68_7"/><b>then </b><i><font color="Green" title="E67">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="E4:68_7"/>
<font color="Maroon" title="c21">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="#E1:68_7"><i><font color="Green" title="E66">A27</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E5:68_7"/><b>then </b>
<font color="Maroon" title="c21">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="#E30:68"><i><font color="Green" title="E63">A24</font></i></a>, <a class="txt" href="#E1:68_7"><i><font color="Green" title="E66">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="E6:68_7"/><b>then </b>
not <font color="Maroon" title="c21">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E7:68_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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:68_7"><i><font color="Green" title="E67">A28</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E66">A29</font></i>:  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">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#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c21">r</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="#E33:68"><i><font color="Green" title="E65">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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
<b>by </b><i><a class="ref" href="vectsp_2.html#T90" title="VECTSP_2:th.90">VECTSP_2:90</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="E67">A30</font></i>:  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">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#K7" title="RLVECT_1:func.7">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#K7" title="RLVECT_1:func.7">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#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="txt" href="#E35:68"><i><font color="Green" title="E66">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#K7" title="RLVECT_1:func.7">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="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/>
<a NAME="E37:68"/><i><font color="Green" title="E68">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="68_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:68_10"/><b>then </b><i><font color="Green" title="E69">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="E4:68_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="#E7:68"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E14:68"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="#E2:68_10"><i><font color="Green" title="E69">A32</font></i></a>, <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="E5:68_10"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E38:68"/><i><font color="Green" title="E69">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="68_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:68_11"/><b>then </b><i><font color="Green" title="E70">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="E4:68_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="#E11:68"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="#E22:68"><i><font color="Green" title="E58">A17</font></i></a>, <a class="txt" href="#E2:68_11"><i><font color="Green" title="E70">A34</font></i></a>, <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="E5:68_11"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E39:68"/>
 <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="68_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:68_12"/><b>then </b><i><font color="Green" title="E70">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="E4:68_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="#E3:68"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E30:68"><i><font color="Green" title="E63">A24</font></i></a>, <a class="txt" href="#E2:68_12"><i><font color="Green" title="E70">A35</font></i></a>, <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="E5:68_12"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E40:68"/><b>then </b><i><font color="Green" title="E70">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="#E2:68"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="#E6:68"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="#E10:68"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="#E15:68"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="#E23:68"><i><font color="Green" title="E59">A18</font></i></a>, <a class="txt" href="#E31:68"><i><font color="Green" title="E64">A25</font></i></a>, <a class="txt" href="#E37:68"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="#E38:68"><i><font color="Green" title="E69">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="E41:68"/>
(  <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="E42:68"/><b>then </b><i><font color="Green" title="E71">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="#E3:68"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E7:68"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="#E11:68"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="#E14:68"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="#E22:68"><i><font color="Green" title="E58">A17</font></i></a>, <a class="txt" href="#E30:68"><i><font color="Green" title="E63">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="E43:68"/>
( <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="E44:68"/><b>then </b><i><font color="Green" title="E72">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="#E42:68"><i><font color="Green" title="E71">A37</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E45:68"/><b>then </b><i><font color="Green" title="E73">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="#E40:68"><i><font color="Green" title="E70">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="E46:68"/><b>then </b><i><font color="Green" title="E74">A40</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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="E48:68"/><i><font color="Green" title="E75">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="E49:68"/><i><font color="Green" title="E76">A42</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">P</font> 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="E50:68"/><i><font color="Green" title="E77">A43</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">P</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> <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="#E48:68"><i><font color="Green" title="E75">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E51:68"/><i><font color="Green" title="E78">A44</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#NK2" title="RELAT_1:NK.2">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="68_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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <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> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> )</span><br/>

<b>assume </b><a NAME="E1:68_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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></span><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="E3:68_13"/><i><font color="Green" title="E79">A45</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">P</font>
 <b>and </b><br/><a NAME="E4:68_13"/><i><font color="Green" title="E80">A46</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="#E3:68_13"><i><font color="Green" title="E79">A45</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:68_13"/><i><font color="Green" title="E81">A47</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="E7:68_13"/><i><font color="Green" title="E82">A48</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#NK2" title="RELAT_1:NK.2">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="#E48:68"><i><font color="Green" title="E75">A41</font></i></a>, <a class="txt" href="#E3:68_13"><i><font color="Green" title="E79">A45</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:68_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="#E44:68"><i><font color="Green" title="E72">A38</font></i></a>, <a class="txt" href="#E46:68"><i><font color="Green" title="E74">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="E9:68_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="#E40:68"><i><font color="Green" title="E70">A36</font></i></a>, <a class="txt" href="#E49:68"><i><font color="Green" title="E76">A42</font></i></a>, <a class="txt" href="#E6:68_13"><i><font color="Green" title="E81">A47</font></i></a>, <a class="txt" href="#E7:68_13"><i><font color="Green" title="E82">A48</font></i></a>, <a class="txt" href="#E50:68"><i><font color="Green" title="E77">A43</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="E10:68_13"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E4:68_13"><i><font color="Green" title="E80">A46</font></i></a>, <a class="ref" href="finseq_4.html#D3" title="FINSEQ_4:def.3">FINSEQ_4:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E79">A49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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> implies <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> ) )</span><br/><b>thus </b><a NAME="E1:68_14"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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> implies <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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> )</span><br/>

<b>hence </b><a NAME="E2:68_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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">P</font>
 <b>by </b><i><a class="txt" href="#E45:68"><i><font color="Green" title="E73">A39</font></i></a>, <a class="txt" href="#E48:68"><i><font color="Green" title="E75">A41</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#NK2" title="RELAT_1:NK.2">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></span><br/>

<a NAME="E3:68_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#NK2" title="RELAT_1:NK.2">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="#E51:68"><i><font color="Green" title="E78">A44</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="E4:68_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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:68_14"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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></span><br/><a NAME="E3:68_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="E4:68_14"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E45:68"><i><font color="Green" title="E73">A39</font></i></a>, <a class="txt" href="#E48:68"><i><font color="Green" title="E75">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E80">A50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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 <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> )</span><br/><b>assume </b><a NAME="E1:68_15"/><i><font color="Green" title="E81">A51</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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></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="E3:68_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="#E44:68"><i><font color="Green" title="E72">A38</font></i></a>, <a class="txt" href="#E1:68_15"><i><font color="Green" title="E81">A51</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="E4:68_15"/><b>then </b><i><font color="Green" title="E82">A52</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="#E40:68"><i><font color="Green" title="E70">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="E5:68_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="#E45:68"><i><font color="Green" title="E73">A39</font></i></a>, <a class="txt" href="#E1:68_15"><i><font color="Green" title="E81">A51</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="#E49:68"><i><font color="Green" title="E76">A42</font></i></a>, <a class="txt" href="#E50:68"><i><font color="Green" title="E77">A43</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="#E4:68_15"><i><font color="Green" title="E82">A52</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="E7:68_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E54:68"/><b>then </b><i><font color="Green" title="E81">A53</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="#E52:68"><i><font color="Green" title="E79">A49</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>
<a NAME="E55:68"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="68_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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> or <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> )</span><br/>

<b>assume </b><a NAME="E1:68_16"/><i><font color="Green" title="E82">A54</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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></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="E2:68_16"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E40:68"><i><font color="Green" title="E70">A36</font></i></a>, <a class="txt" href="#E44:68"><i><font color="Green" title="E72">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="E83">A55</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#NK2" title="RELAT_1:NK.2">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="#E40:68"><i><font color="Green" title="E70">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="#E54:68"><i><font color="Green" title="E81">A53</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="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E40:68"><i><font color="Green" title="E70">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="#E51:68"><i><font color="Green" title="E78">A44</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="E5:68_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="#E1:68_16"><i><font color="Green" title="E82">A54</font></i></a>, <a class="txt" href="#E3:68_16"><i><font color="Green" title="E83">A55</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E56:68"/><b>then </b><i><font color="Green" title="E82">A56</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E51:68"><i><font color="Green" title="E78">A44</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="E57:68"/><i><font color="Green" title="E83">A57</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E48:68"><i><font color="Green" title="E75">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="E58:68"/><b>then </b><i><font color="Green" title="E84">A58</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="#E56:68"><i><font color="Green" title="E82">A56</font></i></a>, <a class="ref" href="finseq_4.html#T75" title="FINSEQ_4:th.75">FINSEQ_4:75</a></i>;<br/>
<a NAME="E59:68"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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="#E51:68"><i><font color="Green" title="E78">A44</font></i></a>, <a class="txt" href="#E57:68"><i><font color="Green" title="E83">A57</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="E61:68"/><i><font color="Green" title="E85">A59</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="E62:68"/><b>then </b><i><font color="Green" title="E86">A60</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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="partfun1.html#K1" title="PARTFUN1:func.1">*</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#NK2" title="RELAT_1:NK.2">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="#E56:68"><i><font color="Green" title="E82">A56</font></i></a>, <a class="txt" href="#E58:68"><i><font color="Green" title="E84">A58</font></i></a>, <a class="txt" href="#E62:68"><i><font color="Green" title="E86">A60</font></i></a>, <a class="ref" href="funct_2.html#T83" title="FUNCT_2:th.83">FUNCT_2:83</a></i>;<br/>
<a NAME="E66:68"/><i><font color="Green" title="E87">A61</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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">P</font>
 
;<br/>
<a NAME="E67:68"/><b>then </b><i><font color="Green" title="E88">A62</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c27">Fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><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="E69:68"/><i><font color="Green" title="E89">A63</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="E70:68"/><i><font color="Green" title="E90">A64</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c29">R</font> 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="E71:68"/><i><font color="Green" title="E91">A65</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c29">R</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> <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="#E69:68"><i><font color="Green" title="E89">A63</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E72:68"/><i><font color="Green" title="E92">A66</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#NK2" title="RELAT_1:NK.2">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="68_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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <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> or <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> )</span><br/>

<b>assume </b><a NAME="E1:68_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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></span><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="E3:68_17"/><i><font color="Green" title="E93">A67</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c29">R</font>
 <b>and </b><br/><a NAME="E4:68_17"/><i><font color="Green" title="E94">A68</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="#E3:68_17"><i><font color="Green" title="E93">A67</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:68_17"/><i><font color="Green" title="E95">A69</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="E7:68_17"/><i><font color="Green" title="E96">A70</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#NK2" title="RELAT_1:NK.2">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="#E69:68"><i><font color="Green" title="E89">A63</font></i></a>, <a class="txt" href="#E3:68_17"><i><font color="Green" title="E93">A67</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:68_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="#E44:68"><i><font color="Green" title="E72">A38</font></i></a>, <a class="txt" href="#E46:68"><i><font color="Green" title="E74">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="E9:68_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="#E40:68"><i><font color="Green" title="E70">A36</font></i></a>, <a class="txt" href="#E70:68"><i><font color="Green" title="E90">A64</font></i></a>, <a class="txt" href="#E6:68_17"><i><font color="Green" title="E95">A69</font></i></a>, <a class="txt" href="#E7:68_17"><i><font color="Green" title="E96">A70</font></i></a>, <a class="txt" href="#E71:68"><i><font color="Green" title="E91">A65</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="E10:68_17"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E4:68_17"><i><font color="Green" title="E94">A68</font></i></a>, <a class="ref" href="finseq_4.html#D3" title="FINSEQ_4:def.3">FINSEQ_4:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E93">A71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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> implies <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> ) )</span><br/><b>thus </b><a NAME="E1:68_18"/>
( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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> implies <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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> )</span><br/>

<a NAME="E2:68_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="#E45:68"><i><font color="Green" title="E73">A39</font></i></a>, <a class="txt" href="#E69:68"><i><font color="Green" title="E89">A63</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="E3:68_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#NK2" title="RELAT_1:NK.2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#NK2" title="RELAT_1:NK.2">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></span><br/>

<a NAME="E4:68_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#NK2" title="RELAT_1:NK.2">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="#E72:68"><i><font color="Green" title="E92">A66</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="E5:68_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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:68_18"/>
( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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></span><br/><a NAME="E3:68_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="E4:68_18"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E45:68"><i><font color="Green" title="E73">A39</font></i></a>, <a class="txt" href="#E69:68"><i><font color="Green" title="E89">A63</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E94">A72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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 <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> )</span><br/><b>assume </b><a NAME="E1:68_19"/><i><font color="Green" title="E95">A73</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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></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="E3:68_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="#E44:68"><i><font color="Green" title="E72">A38</font></i></a>, <a class="txt" href="#E1:68_19"><i><font color="Green" title="E95">A73</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="E4:68_19"/><b>then </b><i><font color="Green" title="E96">A74</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="#E40:68"><i><font color="Green" title="E70">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="E5:68_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="#E45:68"><i><font color="Green" title="E73">A39</font></i></a>, <a class="txt" href="#E1:68_19"><i><font color="Green" title="E95">A73</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="#E70:68"><i><font color="Green" title="E90">A64</font></i></a>, <a class="txt" href="#E71:68"><i><font color="Green" title="E91">A65</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="#E4:68_19"><i><font color="Green" title="E96">A74</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="E7:68_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E75:68"/><b>then </b><i><font color="Green" title="E95">A75</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="#E73:68"><i><font color="Green" title="E93">A71</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>
<a NAME="E76:68"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="68_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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> or <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> )</span><br/>

<b>assume </b><a NAME="E1:68_20"/><i><font color="Green" title="E96">A76</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#NK2" title="RELAT_1:NK.2">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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></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="E2:68_20"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E40:68"><i><font color="Green" title="E70">A36</font></i></a>, <a class="txt" href="#E44:68"><i><font color="Green" title="E72">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="E97">A77</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#NK2" title="RELAT_1:NK.2">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="#E40:68"><i><font color="Green" title="E70">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="#E75:68"><i><font color="Green" title="E95">A75</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="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E40:68"><i><font color="Green" title="E70">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="#E72:68"><i><font color="Green" title="E92">A66</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="E5:68_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="#E1:68_20"><i><font color="Green" title="E96">A76</font></i></a>, <a class="txt" href="#E3:68_20"><i><font color="Green" title="E97">A77</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E77:68"/><b>then </b><i><font color="Green" title="E96">A78</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E72:68"><i><font color="Green" title="E92">A66</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="E78:68"/><i><font color="Green" title="E97">A79</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c29">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="#E69:68"><i><font color="Green" title="E89">A63</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E79:68"/><b>then </b><i><font color="Green" title="E98">A80</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="#E77:68"><i><font color="Green" title="E96">A78</font></i></a>, <a class="ref" href="finseq_4.html#T75" title="FINSEQ_4:th.75">FINSEQ_4:75</a></i>;<br/>
<a NAME="E80:68"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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="#E72:68"><i><font color="Green" title="E92">A66</font></i></a>, <a class="txt" href="#E78:68"><i><font color="Green" title="E97">A79</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="E82:68"/><i><font color="Green" title="E99">A81</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="E83:68"/><b>then </b><i><font color="Green" title="E100">A82</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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="partfun1.html#K1" title="PARTFUN1:func.1">*</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#NK2" title="RELAT_1:NK.2">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="#E77:68"><i><font color="Green" title="E96">A78</font></i></a>, <a class="txt" href="#E79:68"><i><font color="Green" title="E98">A80</font></i></a>, <a class="txt" href="#E83:68"><i><font color="Green" title="E100">A82</font></i></a>, <a class="ref" href="funct_2.html#T83" title="FUNCT_2:th.83">FUNCT_2:83</a></i>;<br/>
<a NAME="E87:68"/><i><font color="Green" title="E101">A83</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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c33">R</font>
 
;<br/>
<a NAME="E88:68"/><b>then </b><i><font color="Green" title="E102">A84</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c32">Hp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">Hp</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="E90:68"/><i><font color="Green" title="E103">A85</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="E91:68"/><i><font color="Green" title="E104">A86</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c34">I</font> 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="E92:68"/><i><font color="Green" title="E105">A87</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> <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="#E90:68"><i><font color="Green" title="E103">A85</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E93:68"/><i><font color="Green" title="E106">A88</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>by </b><i><a class="txt" href="#E91:68"><i><font color="Green" title="E104">A86</font></i></a>, <a class="txt" href="#E92:68"><i><font color="Green" title="E105">A87</font></i></a></i>;<br/>
<a NAME="E94:68"/>
 <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="68_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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <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> or <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> )</span><br/>

<b>assume </b><a NAME="E1:68_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></span><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="E3:68_21"/><i><font color="Green" title="E107">A89</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c34">I</font>
 <b>and </b><br/><a NAME="E4:68_21"/><i><font color="Green" title="E108">A90</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="#E3:68_21"><i><font color="Green" title="E107">A89</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:68_21"/>
<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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">Hp</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E91:68"><i><font color="Green" title="E104">A86</font></i></a>, <a class="txt" href="#E3:68_21"><i><font color="Green" title="E107">A89</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:68_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="#E4:68_21"><i><font color="Green" title="E108">A90</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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="E96:68"/><i><font color="Green" title="E107">A91</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="#E45:68"><i><font color="Green" title="E73">A39</font></i></a>, <a class="txt" href="#E61:68"><i><font color="Green" title="E85">A59</font></i></a>, <a class="txt" href="#E66:68"><i><font color="Green" title="E87">A61</font></i></a>, <a class="txt" href="#E90:68"><i><font color="Green" title="E103">A85</font></i></a>, <a class="ref" href="finseq_2.html#T48" title="FINSEQ_2:th.48">FINSEQ_2:48</a></i>;<br/>
<a NAME="E97:68"/><b>then </b><i><font color="Green" title="E108">A92</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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="E98:68"/><i><font color="Green" title="E109">A93</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="#E45:68"><i><font color="Green" title="E73">A39</font></i></a>, <a class="txt" href="#E82:68"><i><font color="Green" title="E99">A81</font></i></a>, <a class="txt" href="#E87:68"><i><font color="Green" title="E101">A83</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="68_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> )</span><br/><b>assume </b><a NAME="E1:68_22"/><i><font color="Green" title="E110">A94</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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></span><br/><a NAME="E2:68_22"/><b>then </b><i><font color="Green" title="E111">A95</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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c27">Fp</font> )
 <b>by </b><i><a class="txt" href="#E45:68"><i><font color="Green" title="E73">A39</font></i></a>, <a class="txt" href="#E90:68"><i><font color="Green" title="E103">A85</font></i></a>, <a class="txt" href="#E96:68"><i><font color="Green" title="E107">A91</font></i></a>, <a class="txt" href="#E98:68"><i><font color="Green" title="E109">A93</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="#E1:68_22"><i><font color="Green" title="E110">A94</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font>;<br/><i><font color="Green" title="E112">A96</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="#E2:68_22"><i><font color="Green" title="E111">A95</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:68_22"><i><font color="Green" title="E111">A95</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="E113">A97</font></i>: <font color="Maroon" title="c32">Hp</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="partfun1.html#K1" title="PARTFUN1:func.1">*</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="#E2:68_22"><i><font color="Green" title="E111">A95</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</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="#E2:68_22"><i><font color="Green" title="E111">A95</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E6:68_22"/><i><font color="Green" title="E114">A98</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c33">R</font>
 <b>by </b><i><a class="txt" href="#E77:68"><i><font color="Green" title="E96">A78</font></i></a>, <a class="txt" href="#E78:68"><i><font color="Green" title="E97">A79</font></i></a>, <a class="txt" href="#E2:68_22"><i><font color="Green" title="E111">A95</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="E7:68_22"/><i><font color="Green" title="E115">A99</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#NK2" title="RELAT_1:NK.2">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="#E77:68"><i><font color="Green" title="E96">A78</font></i></a>, <a class="txt" href="#E78:68"><i><font color="Green" title="E97">A79</font></i></a>, <a class="txt" href="#E2:68_22"><i><font color="Green" title="E111">A95</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="#E6:68_22"><i><font color="Green" title="E114">A98</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="#E74:68"><i><font color="Green" title="E94">A72</font></i></a>, <a class="txt" href="#E2:68_22"><i><font color="Green" title="E111">A95</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="#E2:68_22"><i><font color="Green" title="E111">A95</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><a NAME="E10:68_22"/><b>then </b><i><font color="Green" title="E116">A100</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:68_22"><i><font color="Green" title="E115">A99</font></i></a>, <a class="ref" href="rmod_4.html#T32" target="_self" title="RMOD_4:th.32">Th32</a></i>;<br/><a NAME="E11:68_22"/><i><font color="Green" title="E117">A101</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">P</font>
 <b>by </b><i><a class="txt" href="#E46:68"><i><font color="Green" title="E74">A40</font></i></a>, <a class="txt" href="#E56:68"><i><font color="Green" title="E82">A56</font></i></a>, <a class="txt" href="#E57:68"><i><font color="Green" title="E83">A57</font></i></a>, <a class="txt" href="#E2:68_22"><i><font color="Green" title="E111">A95</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="E12:68_22"/><i><font color="Green" title="E118">A102</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#NK2" title="RELAT_1:NK.2">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="#E46:68"><i><font color="Green" title="E74">A40</font></i></a>, <a class="txt" href="#E56:68"><i><font color="Green" title="E82">A56</font></i></a>, <a class="txt" href="#E57:68"><i><font color="Green" title="E83">A57</font></i></a>, <a class="txt" href="#E2:68_22"><i><font color="Green" title="E111">A95</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="#E11:68_22"><i><font color="Green" title="E117">A101</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="#E53:68"><i><font color="Green" title="E80">A50</font></i></a>, <a class="txt" href="#E2:68_22"><i><font color="Green" title="E111">A95</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="#E2:68_22"><i><font color="Green" title="E111">A95</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><b>then </b><i><font color="Green" title="E119">A103</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E12:68_22"><i><font color="Green" title="E118">A102</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="vectsp_2.html#K5" title="VECTSP_2:func.5">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E90:68"><i><font color="Green" title="E103">A85</font></i></a>, <a class="txt" href="#E91:68"><i><font color="Green" title="E104">A86</font></i></a>, <a class="txt" href="#E1:68_22"><i><font color="Green" title="E110">A94</font></i></a>, <a class="txt" href="#E4:68_22"><i><font color="Green" title="E112">A96</font></i></a>, <a class="txt" href="#E5:68_22"><i><font color="Green" title="E113">A97</font></i></a>, <a class="txt" href="#E10:68_22"><i><font color="Green" title="E116">A100</font></i></a>, <a class="txt" href="#E92:68"><i><font color="Green" title="E105">A87</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="#E2:68_22"><i><font color="Green" title="E111">A95</font></i></a>, <a class="txt" href="#E15:68_22"><i><font color="Green" title="E119">A103</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E100:68"/><b>then </b><i><font color="Green" title="E110">A104</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="#E97:68"><i><font color="Green" title="E108">A92</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E101:68"/>
 <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#NK2" title="RELAT_1:NK.2">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="#E98:68"><i><font color="Green" title="E109">A93</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="E102:68"/>
 <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#K3" title="RLVECT_1:func.3">+</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="#E4:68"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E8:68"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="#E12:68"><i><font color="Green" title="E52">A9</font></i></a>, <a class="txt" href="#E20:68"><i><font color="Green" title="E57">A16</font></i></a>, <a class="txt" href="#E28:68"><i><font color="Green" title="E62">A23</font></i></a>, <a class="txt" href="#E36:68"><i><font color="Green" title="E67">A30</font></i></a>, <a class="txt" href="#E67:68"><i><font color="Green" title="E88">A62</font></i></a>, <a class="txt" href="#E88:68"><i><font color="Green" title="E102">A84</font></i></a>, <a class="txt" href="#E90:68"><i><font color="Green" title="E103">A85</font></i></a>, <a class="txt" href="#E93:68"><i><font color="Green" title="E106">A88</font></i></a>, <a class="txt" href="#E98:68"><i><font color="Green" title="E109">A93</font></i></a>, <a class="txt" href="#E100:68"><i><font color="Green" title="E110">A104</font></i></a>, <a class="ref" href="rlvect_2.html#T4" title="RLVECT_2:th.4">RLVECT_2:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
