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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</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="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  or  ex <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_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="E1:68_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:68_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  or  ex <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:68_1_1_1"/>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  or  ex <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#T42" target="_self" title="MODAL_1:th.42">Th42</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:68_1_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  or  ex <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:68_1_1_2"/>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  or  ex <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#T43" target="_self" title="MODAL_1:th.43">Th43</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:68"/>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  or  ex <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> or  ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</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/>


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