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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <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">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  holds <br/> <a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Olive" title="b1">A1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">A1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</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="c2">A1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  implies  <a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c2">A1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">A1</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:24"/>
<font color="Maroon" title="c2">A1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c2">A1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">A1</font></span><br/>

<a NAME="E2:24"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="prob_4.html#E19"><span class="lab"><font color="Green" title="E16">Lm2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24"/>
 <a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c2">A1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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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