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<div class="add">

<b>let </b><font color="Maroon" title="c6">X1</font>, <font color="Maroon" title="c7">X2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">DK</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6_1_2"/><i><font color="Green" title="E4">A5</font></i>: 
 ex <font color="Olive" title="b1">p1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">DX</font> st <br/>( <font color="Olive" title="b1">p1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Olive" title="b1">p1</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:6_1_2"/><i><font color="Green" title="E5">A6</font></i>: 
 ex <font color="Olive" title="b1">p2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">DX</font> st <br/>( <font color="Olive" title="b1">p2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Olive" title="b1">p2</font></span>)</span> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">p1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">DX</font><b> such that </b><br/><a NAME="E4:6_1_2"/><i><font color="Green" title="E6">A7</font></i>: 
<font color="Maroon" title="c8">p1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:6_1_2"/><i><font color="Green" title="E7">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font>
 <b>and </b><br/><a NAME="E6:6_1_2"/><i><font color="Green" title="E8">A9</font></i>: 
<font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E1:6_1_2"><i><font color="Green" title="E4">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">p2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">DX</font><b> such that </b><br/><a NAME="E8:6_1_2"/><i><font color="Green" title="E9">A10</font></i>: 
<font color="Maroon" title="c9">p2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E9:6_1_2"/><i><font color="Green" title="E10">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font>
 <b>and </b><br/><a NAME="E10:6_1_2"/><i><font color="Green" title="E11">A12</font></i>: 
<font color="Maroon" title="c7">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c9">p2</font></span>)</span>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E2:6_1_2"><i><font color="Green" title="E5">A6</font></i></a></i>;<br/>
<a NAME="E11:6_1_2"/><i><font color="Green" title="E12">A13</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">p2</font> &amp;  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font> st <br/>( <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">P</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font> ) )
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E4:6_1_2"><i><font color="Green" title="E6">A7</font></i></a>, <a class="txt" href="bhsp_5.html#E5:6_1_2"><i><font color="Green" title="E7">A8</font></i></a>, <a class="txt" href="bhsp_5.html#E8:6_1_2"><i><font color="Green" title="E9">A10</font></i></a>, <a class="txt" href="bhsp_5.html#E9:6_1_2"><i><font color="Green" title="E10">A11</font></i></a>, <a class="ref" href="bhsp_5.html#T1" target="_self" title="BHSP_5:th.1">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">P</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font><b> such that </b><br/><a NAME="E13:6_1_2"/><i><font color="Green" title="E13">A14</font></i>: 
<font color="Maroon" title="c9">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font>
 <b>and </b><br/><a NAME="E14:6_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font>
 <b>and </b><br/><a NAME="E15:6_1_2"/>
 <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>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E4:6_1_2"><i><font color="Green" title="E6">A7</font></i></a>, <a class="txt" href="bhsp_5.html#E5:6_1_2"><i><font color="Green" title="E7">A8</font></i></a>, <a class="txt" href="bhsp_5.html#E8:6_1_2"><i><font color="Green" title="E9">A10</font></i></a>, <a class="txt" href="bhsp_5.html#E9:6_1_2"><i><font color="Green" title="E10">A11</font></i></a>, <a class="ref" href="bhsp_5.html#T1" target="_self" title="BHSP_5:th.1">Th1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">xd</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:6_1_2_1"/>
( <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font> implies <font color="Maroon" title="c8">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xd</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="c8">p1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E2:6_1_2_1"/>
( <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> iff <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E2:6"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="bhsp_5.html#E5:6_1_2"><i><font color="Green" title="E7">A8</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:6_1_2"/><b>then </b><i><font color="Green" title="E14">A15</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">xd</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:6_1_2_2"/>
( <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">p2</font> implies <font color="Maroon" title="c9">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xd</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="c9">p2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E2:6_1_2_2"/>
( <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c9">p2</font></span>)</span> iff <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">p2</font> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E2:6"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="bhsp_5.html#E9:6_1_2"><i><font color="Green" title="E10">A11</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:6_1_2"/><b>then </b><i><font color="Green" title="E15">A16</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c9">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">p2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E20:6_1_2"/><i><font color="Green" title="E16">A17</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> &amp;  <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>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E17:6_1_2"><i><font color="Green" title="E14">A15</font></i></a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E21:6_1_2"/>
 <a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c9">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">xd</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:6_1_2_3_1"/>
( <font color="Maroon" title="c11">xd</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="c10">P</font> implies <font color="Maroon" title="c10">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E20:6_1_2"><i><font color="Green" title="E16">A17</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E2:6_1_2_3_1"/><b>then </b>
( <font color="Maroon" title="c11">xd</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"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font></span>)</span> iff <font color="Maroon" title="c11">xd</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="c10">P</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>hence </b><a NAME="E3:6_1_2_3_1"/>
( <font color="Maroon" title="c11">xd</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"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font></span>)</span> iff <font color="Maroon" title="c11">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c9">p2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E11:6_1_2"><i><font color="Green" title="E12">A13</font></i></a>, <a class="txt" href="bhsp_5.html#E19:6_1_2"><i><font color="Green" title="E15">A16</font></i></a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:6_1_2_3"/><b>then </b><i><font color="Green" title="E17">A18</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c9">p2</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"><span class="p2">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:6_1_2_3_2"/><i><font color="Green" title="E18">A19</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c9">p2</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c11">s</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:6_1_2_3_2"/>
<font color="Maroon" title="c12">i</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="c10">P</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E11:6_1_2"><i><font color="Green" title="E12">A13</font></i></a>, <a class="txt" href="bhsp_5.html#E19:6_1_2"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="bhsp_5.html#E1:6_1_2_3_2"><i><font color="Green" title="E18">A19</font></i></a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/><a NAME="E4:6_1_2_3_2"/><b>then </b><i><font color="Green" title="E19">A20</font></i>: 
<font color="Maroon" title="c10">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</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="c10">P</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E5:6_1_2_3_2"/><b>then </b>
<font color="Maroon" title="c10">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c9">p2</font></span>)</span>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E11:6_1_2"><i><font color="Green" title="E12">A13</font></i></a>, <a class="txt" href="bhsp_5.html#E19:6_1_2"><i><font color="Green" title="E15">A16</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E7:6_1_2_3_2"/><i><font color="Green" title="E20">A21</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E4:6_1_2_3_2"><i><font color="Green" title="E19">A20</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E8:6_1_2_3_2"/><i><font color="Green" title="E21">A22</font></i>: 
<font color="Maroon" title="c11">s</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="c10">P</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E11:6_1_2"><i><font color="Green" title="E12">A13</font></i></a>, <a class="txt" href="bhsp_5.html#E19:6_1_2"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="bhsp_5.html#E1:6_1_2_3_2"><i><font color="Green" title="E18">A19</font></i></a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/><span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c9">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font> = 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E19:6_1_2"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="bhsp_5.html#E1:6_1_2_3_2"><i><font color="Green" title="E18">A19</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E13:6_1_2"><i><font color="Green" title="E13">A14</font></i></a>, <a class="txt" href="bhsp_5.html#E8:6_1_2_3_2"><i><font color="Green" title="E21">A22</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
<b>by </b><i><a class="txt" href="bhsp_5.html#E7:6_1_2_3_2"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
<b>by </b><i><a class="txt" href="bhsp_5.html#E8:6_1_2_3_2"><i><font color="Green" title="E21">A22</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E10:6_1_2_3_2"/>
<span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c9">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1_2_3"/>
 <a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c9">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c8">p1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">P</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E2:6_1_2_3"><i><font color="Green" title="E17">A18</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E22:6_1_2"/>
<font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X2</font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E1:6"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="bhsp_5.html#E6:6_1_2"><i><font color="Green" title="E8">A9</font></i></a>, <a class="txt" href="bhsp_5.html#E10:6_1_2"><i><font color="Green" title="E11">A12</font></i></a>, <a class="txt" href="bhsp_5.html#E17:6_1_2"><i><font color="Green" title="E14">A15</font></i></a>, <a class="ref" href="finsop_1.html#T8" title="FINSOP_1:th.8">FINSOP_1:8</a></i>;<br/>


</div>
