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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">F</font>];<br/>
<a NAME="E1:70_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_1_1"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">F</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E2:70_1_1"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">F</font>]</span><br/>

<a NAME="E3:70_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">F</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">F</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70_1_1_1"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">and </span><br/><a NAME="E2:70_1_1_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">F</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">D</font> = <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<a NAME="E3:70_1_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E1:70_1_1_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="stirl2_1.html#T55" target="_self" onmouseover="rs('stirl2_1/T55')" onmouseout="rh()">Th55</a></span>;<br/>
<a NAME="E4:70_1_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T54" target="_self" onmouseover="rs('stirl2_1/T54')" onmouseout="rh()">Th54</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:70_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E4:70_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">F</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">F</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">F</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E2:70_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">F</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">F</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">n</font> =  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">F</font></span>)</span> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E4:70_1"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
;<br/>
<a NAME="E5:70_1"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">F</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">F</font>]</span><br/>

<a NAME="E2:70_1_2"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:70_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">F</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="relat_1.html#T41" onmouseover="rs('relat_1/T41')" onmouseout="rh()">RELAT_1:41</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E6:70_1"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E5:70_1"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E1:70_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:70_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">F</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
