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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#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" 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#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">m</font> being   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Olive" title="b1">F</font>  st <font color="Olive" title="b2">m</font> is  <a href="measure8.html#V1" title="MEASURE8:attr.1">completely-additive</a>  &amp;  ex <font color="Olive" title="b3">Aseq</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Olive" title="b1">F</font> st <br/>( (  for <font color="Olive" title="b4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <font color="Olive" title="b2">m</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">Aseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">Aseq</font></span>)</span> ) holds <br/> for <font color="Olive" title="b3">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Olive" title="b1">F</font></span>)</span>  st <font color="Olive" title="b3">M</font> <a href="measure8.html#R1" title="MEASURE8:pred.1">is_extension_of</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <span class="p2">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Olive" title="b1">F</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Maroon" title="c2">F</font>  st <font color="Olive" title="b1">m</font> is  <a href="measure8.html#V1" title="MEASURE8:attr.1">completely-additive</a>  &amp;  ex <font color="Olive" title="b2">Aseq</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">F</font> st <br/>( (  for <font color="Olive" title="b3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <font color="Olive" title="b1">m</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Aseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">Aseq</font></span>)</span> ) holds <br/> for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span>  st <font color="Olive" title="b2">M</font> <a href="measure8.html#R1" title="MEASURE8:pred.1">is_extension_of</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <span class="p2">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font> be   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">m</font> is  <a href="measure8.html#V1" title="MEASURE8:attr.1">completely-additive</a>  &amp;  ex <font color="Olive" title="b1">Aseq</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">F</font> st <br/>( (  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <font color="Maroon" title="c3">m</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Aseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">Aseq</font></span>)</span> ) implies  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span>  st <font color="Olive" title="b1">M</font> <a href="measure8.html#R1" title="MEASURE8:pred.1">is_extension_of</a> <font color="Maroon" title="c3">m</font> holds <br/><font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <span class="p2">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span> )</span><br/>





<a NAME="E1:59"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:59"/>
<font color="Maroon" title="c3">m</font> is  <a href="measure8.html#V1" title="MEASURE8:attr.1">completely-additive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  for <font color="Olive" title="b1">Aseq</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">F</font>  holds <br/>(  ex <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st  not <font color="Maroon" title="c3">m</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Aseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">Aseq</font></span>)</span> ) or  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span>  st <font color="Olive" title="b1">M</font> <a href="measure8.html#R1" title="MEASURE8:pred.1">is_extension_of</a> <font color="Maroon" title="c3">m</font> holds <br/><font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <span class="p2">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:59"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c4">M1</font> <a href="measure8.html#R1" title="MEASURE8:pred.1">is_extension_of</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">and </span><br/><a NAME="E5:59"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c4">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <span class="p2">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T33" target="_self" title="MEASURE8:th.33">Th33</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:59"/>
 ex <font color="Olive" title="b1">Aseq</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">F</font> st <br/>( (  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <font color="Maroon" title="c3">m</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Aseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">Aseq</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span>  st <font color="Olive" title="b1">M</font> <a href="measure8.html#R1" title="MEASURE8:pred.1">is_extension_of</a> <font color="Maroon" title="c3">m</font> holds <br/><font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <span class="p2">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">Aseq</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E8:59"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <font color="Maroon" title="c3">m</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Aseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">and </span><br/><a NAME="E9:59"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">Aseq</font></span>)</span>
 ;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">M</font> <a href="measure8.html#R1" title="MEASURE8:pred.1">is_extension_of</a> <font color="Maroon" title="c3">m</font> implies <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <span class="p2">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">Bseq</font> =  <a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c5">Aseq</font> as    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T15" target="_self" title="MEASURE8:th.15">Th15</a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:59"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
<font color="Maroon" title="c6">M</font> <a href="measure8.html#R1" title="MEASURE8:pred.1">is_extension_of</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <span class="p2">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/>

<a NAME="E12:59"/>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <span class="p1">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Maroon" title="c3">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T20" target="_self" title="MEASURE8:th.20">Th20</a></span>;<br/>
<a NAME="E13:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A7</font></span>: 
 <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <span class="p1">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Maroon" title="c3">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></span>;<br/>
<a NAME="E14:59"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
 for <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_1"/><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font></span><br/>

<a NAME="E2:59_1"/><span class="lab"><font color="Green" title="E52">A10</font></span>: 
 for <font color="Olive" title="b1">seq</font> being    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c2">F</font> holds  <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">seq</font>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">seq</font> be    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c9">seq</font>)</span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E53">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">seq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span><br/><a NAME="E1:59_1_1_1"/>
<font color="Maroon" title="c9">seq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/><span class="kw">hence </span><a NAME="E2:59_1_1_1"/>
<font color="Maroon" title="c9">seq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E43">A1</font></span></a></span>; <a class="txt" 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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">y</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">seq</font> implies <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">seq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_1_1_2"/><span class="lab"><font color="Green" title="E54">A12</font></span>: 
<font color="Maroon" title="c11">x</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">and </span><br/><a NAME="E4:59_1_1_2"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
<font color="Maroon" title="c9">seq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/><span class="kw">thus </span><a NAME="E5:59_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_1"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E3:59_1_1_2"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E4:59_1_1_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a></span>; <a class="txt" 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="E3:59_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">seq1</font> = <font color="Maroon" title="c9">seq</font> as   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">y</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">seq1</font> implies <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_1_1_3"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">seq1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_1_1_3"/><span class="lab"><font color="Green" title="E54">A14</font></span>: 
<font color="Maroon" title="c12">x</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">and </span><br/><a NAME="E4:59_1_1_3"/><span class="lab"><font color="Green" title="E55">A15</font></span>: 
<font color="Maroon" title="c10">seq1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/><span class="kw">thus </span><a NAME="E5:59_1_1_3"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_1"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E3:59_1_1_3"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="txt" href="#E4:59_1_1_3"><span class="lab"><font color="Green" title="E55">A15</font></span></a></span>; <a class="txt" 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="E6:59_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A16</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">seq1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">Fseq</font> = <font color="Maroon" title="c10">seq1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/>
<a NAME="E8:59_1_1"/>
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">seq1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D3" target="_self" title="MEASURE8:def.3">Def3</a></span>;<br/>
<a NAME="E9:59_1_1"/><span class="kw">then </span>
(  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c10">seq1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c10">seq1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a>, <a class="ref" href="prob_1.html#T17" title="PROB_1:th.17">PROB_1:17</a></span>;<br/>
<a NAME="E10:59_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A17</font></span>: 
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c10">seq1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="measure1.html#T31" title="MEASURE1:th.31">MEASURE1:31</a></span>;<br/>
<a NAME="E11:59_1_1"/>
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">Fseq</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c9">seq</font>)</span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">n</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">Fseq</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c9">seq</font>)</span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_1_1_4"/><span class="lab"><font color="Green" title="E56">A18</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">Fseq</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c9">seq</font>)</span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">n1</font> = <font color="Maroon" title="c12">n</font> as    <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>  ;<br/>
<a NAME="E3:59_1_1_4"/>
<font color="Maroon" title="c13">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">Fseq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_1_4"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">Fseq</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font> = 
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Fseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c13">n1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="measure8.html#D12" target="_self" title="MEASURE8:def.12">Def12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:59_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">Fseq</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c9">seq</font>)</span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D5" target="_self" title="MEASURE8:def.5">Def5</a></span>; <a class="txt" 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="E12:59_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A19</font></span>: 
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">Fseq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c9">seq</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>;<br/>
<a NAME="E13:59_1_1"/>
 <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c11">Fseq</font> is   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T23" title="MEASURE1:th.23">MEASURE1:23</a></span>;<br/>
<a NAME="E14:59_1_1"/><span class="kw">then </span>
 <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c11">Fseq</font> is    <a href="measure2.html#M1" title="MEASURE2:mode.1">N_Measure_fam</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_1_1"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="ref" href="measure2.html#D1" title="MEASURE2:def.1">MEASURE2:def 1</a></span>;<br/>
<a NAME="E15:59_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c11">Fseq</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">Fseq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure2.html#T11" title="MEASURE2:th.11">MEASURE2:11</a></span>;<br/>
<a NAME="E16:59_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c10">seq1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c9">seq</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59_1_1"><span class="lab"><font color="Green" title="E56">A19</font></span></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:59_1_1"/>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c9">seq</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" 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="E3:59_1"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c8">B</font>) holds <br/><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">r</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c8">B</font>) implies <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_1_2"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c8">B</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r</font></span><br/>

<a NAME="E2:59_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">seq</font> being    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c2">F</font> st <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <span class="p1">(<span class="default"><a href="measure8.html#K2" title="MEASURE8:func.2">vol</a> (<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">seq</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D7" target="_self" title="MEASURE8:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:59_1_2"/>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a></span>; <a class="txt" 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="E4:59_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c8">B</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</a></span>;<br/>
<a NAME="E5:59_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="measure8.html#K5" title="MEASURE8:func.5">Svc</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c8">B</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D4" title="XXREAL_2:def.4">XXREAL_2:def 4</a></span>;<br/>
<a NAME="E6:59_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D8" target="_self" title="MEASURE8:def.8">Def8</a></span>;<br/>
<a NAME="E7:59_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <span class="p2">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E1:59_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="measure4.html#D3" title="MEASURE4:def.3">MEASURE4:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:59_1"/>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E1:59_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" 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="E15:59"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
 for <font color="Olive" title="b1">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  ex <font color="Olive" title="b2">k</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>  st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">B</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">k</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>  st <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_2"/>
 ex <font color="Olive" title="b1">k</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>  st <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> or <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">k</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> <span class="kw"> such that </span><br/><a NAME="E3:59_2"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font>
 ;<br/>
<a NAME="E4:59_2"/><span class="lab"><font color="Green" title="E53">A22</font></span>: 
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="measure8.html#D12" target="_self" title="MEASURE8:def.12">Def12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:59_2"/><span class="lab"><font color="Green" title="E54">A23</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">B9</font> = <font color="Maroon" title="c8">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E7:59_2"/><span class="lab"><font color="Green" title="E55">A24</font></span>: 
( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></span>;<br/>
<a NAME="E8:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="prob_1.html#T6" title="PROB_1:th.6">PROB_1:6</a></span>;<br/>
<a NAME="E9:59_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_2"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="ref" href="measure1.html#T31" title="MEASURE1:th.31">MEASURE1:31</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E10:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A26</font></span>: 
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="measure8.html#T34" target="_self" title="MEASURE8:th.34">Th34</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:59_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E8:59_2"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="ref" href="measure1.html#T30" title="MEASURE1:th.30">MEASURE1:30</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E12:59_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>;<br/>
<a NAME="E13:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A27</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E14:59_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T51" title="SUPINF_2:th.51">SUPINF_2:51</a></span>;<br/>
<a NAME="E15:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A28</font></span>: 
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59_2"><span class="lab"><font color="Green" title="E58">A27</font></span></a>, <a class="txt" href="#E10:59_2"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="ref" href="xxreal_3.html#T28" title="XXREAL_3:th.28">XXREAL_3:28</a></span>;<br/>
<a NAME="E16:59_2"/><span class="lab"><font color="Green" title="E60">A29</font></span>: 
<font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="measure8.html#D12" target="_self" title="MEASURE8:def.12">Def12</a></span>;<br/>
<a NAME="E17:59_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E8:59_2"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="ref" href="measure1.html#T30" title="MEASURE1:th.30">MEASURE1:30</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E18:59_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>;<br/>
<a NAME="E19:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A30</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E20:59_2"/>
<font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_2"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E8:59_2"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="ref" href="measure1.html#T31" title="MEASURE1:th.31">MEASURE1:31</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E21:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A31</font></span>: 
<font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E16:59_2"><span class="lab"><font color="Green" title="E60">A29</font></span></a>, <a class="ref" href="measure8.html#T34" target="_self" title="MEASURE8:th.34">Th34</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E22:59_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T51" title="SUPINF_2:th.51">SUPINF_2:51</a></span>;<br/>
<a NAME="E23:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A32</font></span>: 
<font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:59_2"><span class="lab"><font color="Green" title="E61">A30</font></span></a>, <a class="txt" href="#E21:59_2"><span class="lab"><font color="Green" title="E62">A31</font></span></a>, <a class="ref" href="xxreal_3.html#T28" title="XXREAL_3:th.28">XXREAL_3:28</a></span>;<br/>
<a NAME="E24:59_2"/>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E5:59_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E7:59_2"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="ref" href="prob_1.html#T6" title="PROB_1:th.6">PROB_1:6</a></span>;<br/>
<a NAME="E25:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:59_2"><span class="lab"><font color="Green" title="E60">A29</font></span></a>, <a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="xxreal_3.html#T37" title="XXREAL_3:th.37">XXREAL_3:37</a></span>;<br/>
<a NAME="E26:59_2"/>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">B9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">B9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E5:59_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:59_2"/>
<font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:59_2"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E23:59_2"><span class="lab"><font color="Green" title="E63">A32</font></span></a>, <a class="txt" href="#E25:59_2"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" 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="E16:59"/>
 for <font color="Olive" title="b1">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">B</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:59_3"/><span class="lab"><font color="Green" title="E52">A34</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">B9</font> = <font color="Maroon" title="c8">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:59_3"/><span class="lab"><font color="Green" title="E53">A35</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_3_1"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:59_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">Bseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" target="_self" title="MEASURE8:def.2">Def2</a></span>;<br/>
<a NAME="E3:59_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_3"><span class="lab"><font color="Green" title="E52">A34</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:59_3_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" 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">consider </span><font color="Maroon" title="c10">Cseq</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:59_3"/><span class="lab"><font color="Green" title="E54">A36</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c10">Cseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E3:59_3"><span class="lab"><font color="Green" title="E53">A35</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Cseq</font> = <font color="Maroon" title="c10">Cseq</font> as   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E7:59_3"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c11">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">m</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> implies <font color="Maroon" title="c11">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">m</font> )</span><br/>

<a NAME="E1:59_3_2"/><span class="lab"><font color="Green" title="E55">A37</font></span>: 
<font color="Maroon" title="c7">Bseq</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_3.html#T11" title="PROB_3:th.11">PROB_3:11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:59_3_2"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">m</font></span><br/>

<a NAME="E3:59_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c13">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_3_2"><span class="lab"><font color="Green" title="E55">A37</font></span></a>, <a class="ref" href="prob_1.html#D5" title="PROB_1:def.5">PROB_1:def 5</a></span>;<br/>
<a NAME="E4:59_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/>
<a NAME="E5:59_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_3"><span class="lab"><font color="Green" title="E54">A36</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:59_3_2"/>
<font color="Maroon" title="c11">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_3"><span class="lab"><font color="Green" title="E54">A36</font></span></a></span>; <a class="txt" 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="E8:59_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A38</font></span>: 
<font color="Maroon" title="c11">Cseq</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_1.html#D5" title="PROB_1:def.5">PROB_1:def 5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">y</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">Cseq</font> implies <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_3_3"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">Cseq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_3_3"/><span class="lab"><font color="Green" title="E56">A39</font></span>: 
<font color="Maroon" title="c13">x</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">and </span><br/><a NAME="E4:59_3_3"/><span class="lab"><font color="Green" title="E57">A40</font></span>: 
<font color="Maroon" title="c11">Cseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">x</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_3_3"><span class="lab"><font color="Green" title="E56">A39</font></span></a></span>;<br/><a NAME="E6:59_3_3"/>
<font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/><a NAME="E7:59_3_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E1:59_3"><span class="lab"><font color="Green" title="E52">A34</font></span></a>, <a class="ref" href="measure1.html#T34" title="MEASURE1:th.34">MEASURE1:34</a></span>;<br/><span class="kw">hence </span><a NAME="E8:59_3_3"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_3"><span class="lab"><font color="Green" title="E54">A36</font></span></a>, <a class="txt" href="#E4:59_3_3"><span class="lab"><font color="Green" title="E57">A40</font></span></a></span>; <a class="txt" 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="E10:59_3"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">Cseq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">Cseq</font> = <font color="Maroon" title="c11">Cseq</font> as   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E12:59_3"/>
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">n</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_3_4"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">n1</font> = <font color="Maroon" title="c13">n</font> as    <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>  ;<br/>
<div><span class="lab"><font color="Green" title="E56">A41</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n1</font> implies  ex <font color="Olive" title="b1">k</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>  st <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n1</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b1">k</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:59_3_4_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</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>  st <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n1</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b1">k</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">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="E3:59_3_4_1"/><span class="lab"><font color="Green" title="E57">A42</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n1</font>
 <span class="kw">and </span><br/><a NAME="E4:59_3_4_1"/><span class="lab"><font color="Green" title="E58">A43</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T13" title="PROB_3:th.13">PROB_3:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">k</font> = <font color="Maroon" title="c16">k</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c17">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n1</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c18">k</font> )</span><br/><a NAME="E6:59_3_4_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c18">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_3"><span class="lab"><font color="Green" title="E54">A36</font></span></a>, <a class="txt" href="#E4:59_3_4_1"><span class="lab"><font color="Green" title="E58">A43</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:59_3_4_1"/>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n1</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c18">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_3_4_1"><span class="lab"><font color="Green" title="E57">A42</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:59_3_4"/><span class="lab"><font color="Green" title="E57">AA</font></span>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c14">n1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n1</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c14">n1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_3_4_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c14">n1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">k</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> <span class="kw"> such that </span><br/><a NAME="E3:59_3_4_2"/><span class="lab"><font color="Green" title="E58">A44</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n1</font>
 <span class="kw">and </span><br/><a NAME="E4:59_3_4_2"/><span class="lab"><font color="Green" title="E59">A45</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_3_4"><span class="lab"><font color="Green" title="E56">A41</font></span></a></span>;<br/>
<a NAME="E5:59_3_4_2"/>
<font color="Maroon" title="c7">Bseq</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_3.html#T11" title="PROB_3:th.11">PROB_3:11</a></span>;<br/>
<a NAME="E6:59_3_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c14">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_3_4_2"><span class="lab"><font color="Green" title="E58">A44</font></span></a>, <a class="ref" href="prob_1.html#D5" title="PROB_1:def.5">PROB_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:59_3_4_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c14">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_3_4_2"><span class="lab"><font color="Green" title="E59">A45</font></span></a></span>; <a class="txt" 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="E5:59_3_4"/><span class="lab"><font color="Green" title="E58">XX</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/>
<a NAME="E6:59_3_4"/><span class="lab"><font color="Green" title="E59">YY</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E7:59_3_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E8:59_3_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<a NAME="E9:59_3_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_3_4"><span class="lab"><font color="Green" title="E58">XX</font></span></a>, <a class="txt" href="#E15:59"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E4:59_3_4"><span class="lab"><font color="Green" title="E57">AA</font></span></a>, <a class="txt" href="#E6:59_3_4"><span class="lab"><font color="Green" title="E59">YY</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:59_3_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>; <a class="txt" 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="E13:59_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A46</font></span>: 
<font color="Maroon" title="c4">M1</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>;<br/>
<a NAME="E14:59_3"/>
 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  <font color="Maroon" title="c12">Cseq</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> <font color="Maroon" title="c9">B9</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Bseq</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_3"><span class="lab"><font color="Green" title="E54">A36</font></span></a></span>;<br/>
<a NAME="E15:59_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">Cseq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">B9</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c7">Bseq</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D5" title="SETLIM_2:def.5">SETLIM_2:def 5</a></span>;<br/>
<a NAME="E16:59_3"/><span class="kw">then </span>
 <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c12">Cseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B9</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c7">Bseq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#T38" title="SETLIM_2:th.38">SETLIM_2:38</a></span>;<br/>
<a NAME="E17:59_3"/><span class="kw">then </span>
 <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c12">Cseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B9</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c5">Aseq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T15" title="PROB_3:th.15">PROB_3:15</a></span>;<br/>
<a NAME="E18:59_3"/><span class="kw">then </span>
 <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c12">Cseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B9</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>;<br/>
<a NAME="E19:59_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A47</font></span>: 
<font color="Maroon" title="c9">B9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c12">Cseq</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E20:59_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">B9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c12">Cseq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_3"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="ref" href="setlim_1.html#T63" title="SETLIM_1:th.63">SETLIM_1:63</a></span>;<br/>
<a NAME="E21:59_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M1</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c12">Cseq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_3"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="ref" href="measure8.html#T26" target="_self" title="MEASURE8:th.26">Th26</a></span>;<br/>
<a NAME="E22:59_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c12">Cseq</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_3"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E13:59_3"><span class="lab"><font color="Green" title="E56">A46</font></span></a>, <a class="ref" href="prob_4.html#T19" title="PROB_4:th.19">PROB_4:19</a></span>;<br/>
<a NAME="E23:59_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c12">Cseq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_3"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="ref" href="prob_4.html#T19" title="PROB_4:th.19">PROB_4:19</a></span>;<br/>
<a NAME="E24:59_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c12">Cseq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_3"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="ref" href="measure8.html#T26" target="_self" title="MEASURE8:th.26">Th26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:59_3"/>
<font color="Maroon" title="c4">M1</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_3"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E19:59_3"><span class="lab"><font color="Green" title="E57">A47</font></span></a>, <a class="ref" href="setlim_1.html#T63" title="SETLIM_1:th.63">SETLIM_1:63</a></span>; <a class="txt" 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="E17:59"/>
<font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <span class="p2">(<span class="default"><a href="measure8.html#K9" title="MEASURE8:func.9">C_Meas</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
