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<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <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> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font>;<br/>
<a NAME="E1:26_1_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_1"/>
( <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">f</font> or  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font> is <font color="Maroon" title="c1">D</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">k</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">n</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/><a NAME="E3:26_1_1_1_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_1_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="afinsq_2.html#T7" target="_self" title="AFINSQ_2:th.7">Th16</a></span>;<br/><a NAME="E4:26_1_1_1_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_1_1_1_1_1"/>
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">m</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:26_1_1_1_1_1"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:26_1_1_1_1_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T3" target="_self" title="AFINSQ_2:th.3">Th3</a></span>;<br/>
<a NAME="E5:26_1_1_1_1_1"/>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_1_1_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E6:26_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E7:26_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<font color="Maroon" title="c7">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E8:26_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_1_1_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="afinsq_2.html#D2" target="_self" title="AFINSQ_2:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:26_1_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_1_1_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E7:26_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:26_1_1_1_1"/>
 for <font color="Olive" title="b1">f2</font> being   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> holds <br/><font color="Olive" title="b1">f2</font> is   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><span class="kw">hence </span><a NAME="E6:26_1_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font> is <font color="Maroon" title="c1">D</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E4:26_1_1_1_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font> is <font color="Maroon" title="c1">D</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> </span><br/><div class="add"><a NAME="E2:26_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T6" target="_self" title="AFINSQ_2:th.6">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_1_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c3">n</font> is <font color="Maroon" title="c1">D</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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