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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">O1</font>, <font color="Maroon" title="c4">O2</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c2">Fx</font> is  <a href="topdim_2.html#V1" title="TOPDIM_2:attr.1">finite-order</a>  &amp; ( for <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c3">O1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> &amp; <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp;  ex <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">O1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 &amp; (  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) &amp; ( for <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> &amp; <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp;  ex <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 &amp; (  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) implies <font color="Maroon" title="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> ) &amp; ( not <font color="Maroon" title="c2">Fx</font> is  <a href="topdim_2.html#V1" title="TOPDIM_2:attr.1">finite-order</a>  &amp; <font color="Maroon" title="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  implies <font color="Maroon" title="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:3_1_2_1"/>
<font color="Maroon" title="c2">Fx</font> is  <a href="topdim_2.html#V1" title="TOPDIM_2:attr.1">finite-order</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c3">O1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> &amp; <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp;  ex <font color="Olive" title="b1">G1</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">O1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 &amp; (  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">G1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Olive" title="b1">G1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) &amp; ( for <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> &amp; <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp;  ex <font color="Olive" title="b1">G2</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 &amp; (  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">G2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Olive" title="b1">G2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) implies <font color="Maroon" title="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> )</span><br/><span class="kw">assume </span><a NAME="E2:3_1_2_1"/><span class="lab"><font color="Green" title="E2">A14</font></span>: 
for <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c3">O1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> &amp; <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <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">G1</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">O1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 &amp; (  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">G1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Olive" title="b1">G1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) &amp; ( for <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> &amp; <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp;  ex <font color="Olive" title="b1">G2</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 &amp; (  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">G2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Olive" title="b1">G2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) implies <font color="Maroon" title="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> )</span><br/><span class="kw">given </span><font color="Maroon" title="c5">G1</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><a NAME="E4:3_1_2_1"/><span class="lab"><font color="Green" title="E3">A15</font></span>: 
<font color="Maroon" title="c5">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font>
 <span class="kw">and </span><br/><a NAME="E5:3_1_2_1"/><span class="lab"><font color="Green" title="E4">A16</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">O1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E6:3_1_2_1"/><span class="lab"><font color="Green" title="E5">A17</font></span>: 
(  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">G1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Maroon" title="c5">G1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> &amp; <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp;  ex <font color="Olive" title="b1">G2</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 &amp; (  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">G2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Olive" title="b1">G2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) implies <font color="Maroon" title="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> )</span><br/><span class="kw">assume </span><a NAME="E7:3_1_2_1"/><span class="lab"><font color="Green" title="E6">A18</font></span>: 
for <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> &amp; <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <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">G2</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 &amp; (  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">G2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Olive" title="b1">G2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) implies <font color="Maroon" title="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> )</span><br/><span class="kw">given </span><font color="Maroon" title="c6">G2</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><a NAME="E9:3_1_2_1"/><span class="lab"><font color="Green" title="E7">A19</font></span>: 
<font color="Maroon" title="c6">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font>
 <span class="kw">and </span><br/><a NAME="E10:3_1_2_1"/><span class="lab"><font color="Green" title="E8">A20</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E11:3_1_2_1"/><span class="lab"><font color="Green" title="E9">A21</font></span>: 
(  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">G2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Maroon" title="c6">G2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font></span><br/><a NAME="E12:3_1_2_1"/>
not  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">G2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_2_1"><span class="lab"><font color="Green" title="E3">A15</font></span></a>, <a class="txt" href="#E6:3_1_2_1"><span class="lab"><font color="Green" title="E5">A17</font></span></a>, <a class="txt" href="#E7:3_1_2_1"><span class="lab"><font color="Green" title="E6">A18</font></span></a>, <a class="txt" href="#E10:3_1_2_1"><span class="lab"><font color="Green" title="E8">A20</font></span></a></span>;<br/><a NAME="E13:3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A22</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">G1</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="Maroon" title="c6">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></span>;<br/><a NAME="E14:3_1_2_1"/>
not  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_2_1"><span class="lab"><font color="Green" title="E2">A14</font></span></a>, <a class="txt" href="#E5:3_1_2_1"><span class="lab"><font color="Green" title="E4">A16</font></span></a>, <a class="txt" href="#E9:3_1_2_1"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="txt" href="#E11:3_1_2_1"><span class="lab"><font color="Green" title="E9">A21</font></span></a></span>;<br/><a NAME="E15:3_1_2_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">G1</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="c6">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:3_1_2_1"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E16:3_1_2_1"/>
<font color="Maroon" title="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3_1_2_1"><span class="lab"><font color="Green" title="E4">A16</font></span></a>, <a class="txt" href="#E10:3_1_2_1"><span class="lab"><font color="Green" title="E8">A20</font></span></a>, <a class="ref" href="xxreal_3.html#T11" title="XXREAL_3:th.11">XXREAL_3:11</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>
<span class="kw">hence </span><a NAME="E2:3_1_2"/>
( ( <font color="Maroon" title="c2">Fx</font> is  <a href="topdim_2.html#V1" title="TOPDIM_2:attr.1">finite-order</a>  &amp; ( for <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c3">O1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> &amp; <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp;  ex <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">O1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 &amp; (  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) &amp; ( for <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> &amp; <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp;  ex <font color="Olive" title="b1">Gx</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Gx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Fx</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">Gx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 &amp; (  not  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Olive" title="b1">Gx</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) implies <font color="Maroon" title="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</font> ) &amp; ( not <font color="Maroon" title="c2">Fx</font> is  <a href="topdim_2.html#V1" title="TOPDIM_2:attr.1">finite-order</a>  &amp; <font color="Maroon" title="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  implies <font color="Maroon" title="c3">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">O2</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>
