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<span class="kw">let </span><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</font>, <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F1</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X1</font><br/>  for <font color="Olive" title="b2">F2</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b3">F</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <font color="Olive" title="b1">F1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">F2</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b2">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b3">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) holds <br/> <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">F1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F2</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b2">F</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <font color="Maroon" title="c5">F1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">F2</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b2">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) holds <br/> <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">F2</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">X2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <font color="Maroon" title="c5">F1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">F2</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) holds <br/> <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">F</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">F1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">F2</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) implies  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c5">F1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E3:23"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c6">F2</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 <span class="kw">and </span><br/><a NAME="E4:23"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E5:23"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><a NAME="E1:23_1"/>
( <font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E20">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:23_1"/>
<font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="measur11.html#T10" target="_self" onmouseover="rs('measur11/T10')" onmouseout="rh()">Th4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:23"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_2.html#T7" onmouseover="rs('prob_2/T7')" onmouseout="rh()">PROB_2:7</a></span>;<br/>
<a NAME="E8:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c7">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c7">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#T63" onmouseover="rs('setlim_1/T63')" onmouseout="rh()">SETLIM_1:63</a></span>;<br/>
<a NAME="E9:23"/>
(  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="setlim_1.html#T63" onmouseover="rs('setlim_1/T63')" onmouseout="rh()">SETLIM_1:63</a></span>;<br/>
<a NAME="E10:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A8</font></span>: 
(  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c5">F1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<a NAME="E11:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> where <font color="Olive" title="b1">P</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c5">F1</font>, <font color="Olive" title="b2">Q</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c6">F2</font> : ( <font color="Olive" title="b1">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c5">F1</font> &amp; <font color="Olive" title="b2">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c6">F2</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#T2" onmouseover="rs('lattice5/T2')" onmouseout="rh()">LATTICE5:2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:23_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:23_2"/><span class="lab"><font color="Green" title="E24">X1</font></span>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c9">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c5">F1</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c6">F2</font> : ( <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c5">F1</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c6">F2</font> ) </span> } </span>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c5">F1</font>, <font color="Maroon" title="c11">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c6">F2</font><span class="kw"> such that </span><br/><a NAME="E5:23_2"/><span class="lab"><font color="Green" title="E25">X2</font></span>: 
( <font color="Maroon" title="c9">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c10">A</font>,<font color="Maroon" title="c11">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <font color="Maroon" title="c10">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c5">F1</font> &amp; <font color="Maroon" title="c11">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c6">F2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_2"><span class="lab"><font color="Green" title="E24">X1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">n1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:23_2"/><span class="lab"><font color="Green" title="E26">X3</font></span>: 
( <font color="Maroon" title="c12">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">F1</font> &amp; <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" onmouseover="rs('partfun1/T3')" onmouseout="rh()">PARTFUN1:3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">n2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E9:23_2"/><span class="lab"><font color="Green" title="E27">X4</font></span>: 
( <font color="Maroon" title="c13">n2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">F2</font> &amp; <font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">n2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" onmouseover="rs('partfun1/T3')" onmouseout="rh()">PARTFUN1:3</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">n</font> =  <a href="nat_1.html#K5" title="NAT_1:func.5">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>);<br/><a NAME="E10:23_2"/>
( <font color="Maroon" title="c10">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span> &amp; <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E7:23_2"><span class="lab"><font color="Green" title="E26">X3</font></span></a>, <a class="txt" href="#E9:23_2"><span class="lab"><font color="Green" title="E27">X4</font></span></a>, <a class="ref" href="prob_1.html#D5" onmouseover="rs('prob_1/D5')" onmouseout="rh()">PROB_1:def 5</a>, <a class="ref" href="xxreal_0.html#T25" onmouseover="rs('xxreal_0/T25')" onmouseout="rh()">XXREAL_0:25</a></span>;<br/><a NAME="E11:23_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">X5</font></span>: 
<font color="Maroon" title="c9">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_2"><span class="lab"><font color="Green" title="E25">X2</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" onmouseover="rs('zfmisc_1/T96')" onmouseout="rh()">ZFMISC_1:96</a></span>;<br/><a NAME="E12:23_2"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 ;<br/><a NAME="E13:23_2"/><span class="kw">then </span>
 <a href="nat_1.html#K5" title="NAT_1:func.5">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E14:23_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E15:23_2"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">max</a> (<font color="Maroon" title="c12">n1</font>,<font color="Maroon" title="c13">n2</font>)</span>)</span></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E20">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:23_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_2"><span class="lab"><font color="Green" title="E24">X1</font></span></a>, <a class="txt" href="#E11:23_2"><span class="lab"><font color="Green" title="E28">X5</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E13:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">X6</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span> implies <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:23_3"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:23_3"/><span class="lab"><font color="Green" title="E25">Y1</font></span>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c9">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:23_3"/><span class="lab"><font color="Green" title="E26">Y2</font></span>: 
( <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c9">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c10">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_3"><span class="lab"><font color="Green" title="E25">Y1</font></span></a>, <a class="ref" href="partfun1.html#T3" onmouseover="rs('partfun1/T3')" onmouseout="rh()">PARTFUN1:3</a></span>;<br/><a NAME="E6:23_3"/><span class="lab"><font color="Green" title="E27">Y3</font></span>: 
<font color="Maroon" title="c9">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E5:23_3"><span class="lab"><font color="Green" title="E26">Y2</font></span></a></span>;<br/><a NAME="E7:23_3"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E8:23_3"/><span class="kw">then </span>
( <font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c5">F1</font></span>)</span> &amp; <font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c6">F2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><a NAME="E9:23_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E6:23_3"><span class="lab"><font color="Green" title="E27">Y3</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" onmouseover="rs('zfmisc_1/T96')" onmouseout="rh()">ZFMISC_1:96</a></span>;<br/><span class="kw">hence </span><a NAME="E10:23_3"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_3"><span class="lab"><font color="Green" title="E25">Y1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E15:23"/><span class="kw">then </span>
 <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<span class="kw">hence </span><a NAME="E16:23"/>
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E24">X6</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
