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<span class="kw">consider </span><font color="Maroon" title="c3">B0</font> being   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c2">x</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">B</font> being  <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c2">x</font> st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B</font>;<br/>
<a NAME="E2:3_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">B0</font> is  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a> 
 
;<br/>
<a NAME="E3:3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">A</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E5:3_1_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">A</font>]
 <span class="kw">and </span><br/><a NAME="E6:3_1_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c4">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="#E3:3_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c5">B1</font> being   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c2">x</font><span class="kw"> such that </span><br/><a NAME="E8:3_1_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">B1</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">B</font> being  <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c2">x</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c2">x</font> holds   <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E9:3_1_1"/>
(  ex <font color="Olive" title="b1">B</font> being  <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c2">x</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c2">x</font> holds   <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">B</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:3_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E8:3_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></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/>


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