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<span class="kw">let </span><font color="Maroon" title="c1">Omega</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Sigma</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font><br/> for <font color="Olive" title="b2">ASeq</font> being  <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b1">Sigma</font>  st <font color="Olive" title="b2">ASeq</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  holds <br/>for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">m</font> holds <br/><span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Olive" title="b2">ASeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Sigma</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font>; <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">ASeq</font> being  <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>  st <font color="Olive" title="b1">ASeq</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  holds <br/>for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Olive" title="b1">ASeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">ASeq</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>; <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">ASeq</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  implies for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c3">ASeq</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</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">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> implies <span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:38"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font></span><br/>

<span class="kw">assume </span><a NAME="E3:38"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:38"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">and </span><br/><a NAME="E6:38"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:38"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">and </span><br/><a NAME="E9:38"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="prob_3.html#T15" target="_self" title="PROB_3:th.15">Th15</a></span>;<br/>
<a NAME="E10:38"/>
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E11:38"/><span class="kw">then </span>
<font color="Maroon" title="c3">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E8:38"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="prob_2.html#D4" title="PROB_2:def.4">PROB_2:def 4</a></span>;<br/>
<a NAME="E12:38"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:38"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:38"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E9:38"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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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