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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">Y</font>, <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">E</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">X</font>,<font color="Olive" title="b1">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><br/>  for <font color="Olive" title="b4">G</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="b3">E</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b4">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Olive" title="b2">x</font>) ) holds <br/><font color="Olive" title="b4">G</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">x</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">E</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">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><br/>  for <font color="Olive" title="b2">G</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">E</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &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">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Maroon" title="c3">x</font>) ) holds <br/><font color="Olive" title="b2">G</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">E</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">X</font>,<font color="Maroon" title="c2">Y</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">  for <font color="Olive" title="b1">G</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="Maroon" title="c4">E</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &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">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Maroon" title="c3">x</font>) ) holds <br/><font color="Olive" title="b1">G</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">G</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" 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="c4">E</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &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="c5">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c3">x</font>) ) implies <font color="Maroon" title="c5">G</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c4">E</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E36">A2</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="c5">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c3">x</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"> <font color="Maroon" title="c5">G</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> </span><br/>

<a NAME="E3:48"/>
 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="c5">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c5">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/>

<a NAME="E1:48_1"/>
 <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c3">x</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T21" target="_self" onmouseover="rs('measur11/T21')" onmouseout="rh()">Th15</a>, <a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="kurato_0.html#D4" onmouseover="rs('kurato_0/D4')" onmouseout="rh()">KURATO_0:def 4</a></span>;<br/>
<a NAME="E2:48_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E36">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:48_1"/>
<font color="Maroon" title="c5">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="#E2:48"><span class="lab"><font color="Green" title="E36">A2</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>
<span class="kw">hence </span><a NAME="E4:48"/>
<font color="Maroon" title="c5">G</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="kurato_0.html#D4" onmouseover="rs('kurato_0/D4')" onmouseout="rh()">KURATO_0: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>
