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<div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">p</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">p</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">p</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">p</font></span>)</span> ) &amp;  ex <font color="Olive" title="b1">p</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">p</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">p</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">p</font></span>)</span> ) implies <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:6_1_2"/><span class="lab"><font color="Green" title="E5">A6</font></span>: 
 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> )
 <span class="kw">and </span><br/><a NAME="E2:6_1_2"/><span class="lab"><font color="Green" title="E6">A7</font></span>: 
 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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X2</font></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:6_1_2"/><span class="lab"><font color="Green" title="E7">A8</font></span>: 
<font color="Maroon" title="c8">p1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E5:6_1_2"/><span class="lab"><font color="Green" title="E8">A9</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E6:6_1_2"/><span class="lab"><font color="Green" title="E9">A10</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_2"><span class="lab"><font color="Green" title="E5">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E8:6_1_2"/><span class="lab"><font color="Green" title="E10">A11</font></span>: 
<font color="Maroon" title="c9">p2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E9:6_1_2"/><span class="lab"><font color="Green" title="E11">A12</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E10:6_1_2"/><span class="lab"><font color="Green" title="E12">A13</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_1_2"><span class="lab"><font color="Green" title="E6">A7</font></span></a></span>;<br/>
<a NAME="E11:6_1_2"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_2"><span class="lab"><font color="Green" title="E7">A8</font></span></a>, <a class="txt" href="#E5:6_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E8:6_1_2"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="txt" href="#E9:6_1_2"><span class="lab"><font color="Green" title="E11">A12</font></span></a>, <a class="ref" href="bhsp_5.html#T1" target="_self" title="BHSP_5:th.1">Th1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">P</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E13:6_1_2"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
<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>
 <span class="kw">and </span><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>
 <span class="kw">and </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_2"><span class="lab"><font color="Green" title="E7">A8</font></span></a>, <a class="txt" href="#E5:6_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E8:6_1_2"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="txt" href="#E9:6_1_2"><span class="lab"><font color="Green" title="E11">A12</font></span></a>, <a class="ref" href="bhsp_5.html#T1" target="_self" title="BHSP_5:th.1">Th1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">xd</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E5:6_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E17:6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A16</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">xd</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E9:6_1_2"><span class="lab"><font color="Green" title="E11">A12</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E19:6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A17</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/>
<a NAME="E20:6_1_2"/><span class="lab"><font color="Green" title="E17">A18</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:6_1_2"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">xd</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/><a NAME="E1:6_1_2_3"/>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:6_1_2"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E2:6_1_2_3"/><span class="kw">then </span>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E3:6_1_2_3"/>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:6_1_2"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E19:6_1_2"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E22:6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A19</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> )</span><br/><span class="kw">assume </span><a NAME="E1:6_1_2_4"/><span class="lab"><font color="Green" title="E19">A20</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><span class="kw">then </span><span class="kw">reconsider </span><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_4"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:6_1_2"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E19:6_1_2"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="txt" href="#E1:6_1_2_4"><span class="lab"><font color="Green" title="E19">A20</font></span></a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></span>;<br/><a NAME="E4:6_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A21</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E5:6_1_2_4"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:6_1_2"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E19:6_1_2"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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_4"/><span class="lab"><font color="Green" title="E21">A22</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_2_4"><span class="lab"><font color="Green" title="E20">A21</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><a NAME="E8:6_1_2_4"/><span class="lab"><font color="Green" title="E22">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:6_1_2"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E19:6_1_2"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="txt" href="#E1:6_1_2_4"><span class="lab"><font color="Green" title="E19">A20</font></span></a>, <a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></span>;<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:6_1_2"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="txt" href="#E1:6_1_2_4"><span class="lab"><font color="Green" title="E19">A20</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:6_1_2"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="txt" href="#E8:6_1_2_4"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:6_1_2_4"><span class="lab"><font color="Green" title="E21">A22</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6_1_2_4"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:6_1_2_4"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E24:6_1_2"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:6_1_2"><span class="lab"><font color="Green" title="E18">A19</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:6_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E6:6_1_2"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="txt" href="#E10:6_1_2"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E17:6_1_2"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="ref" href="finsop_1.html#T8" title="FINSOP_1:th.8">FINSOP_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
