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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p3</font>, <font color="Olive" title="b3">p2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p3</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p3</font> holds <br/><span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b2">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Olive" title="b2">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b2">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p3</font>, <font color="Maroon" title="c4">p2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> implies <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )</span><br/>







<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E49">A1</font></i>: 
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><br/>

<a NAME="E2:55"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="finseq_6.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E3:55"/><b>then </b>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<b>hence </b><span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p3</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">-|</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p3</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_6.html#T44" target="_self" title="FINSEQ_6:th.44">Th44</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p3</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="finseq_6.html#T32" target="_self" title="FINSEQ_6:th.32">Th32</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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