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<span class="kw">let </span><font color="Maroon" title="c4">g1</font>, <font color="Maroon" title="c5">g2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="integra1.html#K1" title="INTEGRA1:func.1">divs</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K1" title="INTEGRA1:func.1">divs</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">D</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="c4">g1</font> holds <br/><font color="Maroon" title="c4">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D</font> ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K1" title="INTEGRA1:func.1">divs</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">D</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="c5">g2</font> holds <br/><font color="Maroon" title="c5">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D</font> ) implies <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:25_2_2"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K1" title="INTEGRA1:func.1">divs</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E2:25_2_2"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
for <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">D</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="c4">g1</font> holds <br/><font color="Maroon" title="c4">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D</font>
 <span class="kw">and </span><br/><a NAME="E3:25_2_2"/><span class="lab"><font color="Green" title="E18">A11</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K1" title="INTEGRA1:func.1">divs</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E4:25_2_2"/><span class="lab"><font color="Green" title="E19">A12</font></span>: 
for <font color="Olive" title="b1">D</font> being   <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">D</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="c5">g2</font> holds <br/><font color="Maroon" title="c5">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">D</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="integra1.html#K1" title="INTEGRA1:func.1">divs</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">D</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="c4">g1</font> implies <font color="Maroon" title="c4">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">D</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">D1</font> = <font color="Maroon" title="c6">D</font> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D3" target="_self" title="INTEGRA1:def.3">Def3</a></span>;<br/><span class="kw">assume </span><a NAME="E2:25_2_2_1"/>
<font color="Maroon" title="c6">D</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="c4">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">D</font></span><br/><a NAME="E3:25_2_2_1"/>
<font color="Maroon" title="c4">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_2_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E2:25_2_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:25_2_2_1"/>
<font color="Maroon" title="c4">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="txt" href="#E4:25_2_2"><span class="lab"><font color="Green" title="E19">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:25_2_2"/>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_2_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E3:25_2_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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