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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b2">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</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"> (  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:61"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">a9</font> = <font color="Maroon" title="c3">a</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">now </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">  for <font color="Olive" title="b1">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> )</span><br/><span class="kw">assume </span><a NAME="E1:61_1"/>
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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"> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font></span><br/><a NAME="E2:61_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:61_1"/>
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="bagorder.html#D13" target="_self" onmouseover="rs('bagorder/D13')" onmouseout="rh()">Def13</a>, <a class="ref" href="card_1.html#T27" onmouseover="rs('card_1/T27')" onmouseout="rh()">CARD_1:27</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="E4:61"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">a</font>
 
;<br/>
<a NAME="E5:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a9</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">a9</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T44" onmouseover="rs('card_2/T44')" onmouseout="rh()">CARD_2:44</a></span>;<br/>
<a NAME="E6:61"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:61"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="bagorder.html#K11" title="BAGORDER:func.11">PosetMax</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E37">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/>


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