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<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I1</font> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I1</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:24_1_1_1"/>
<font color="Maroon" title="c2">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">IT'</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I1</font> ex <font color="Olive" title="b2">I2'</font> being 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>  ex <font color="Olive" title="b3">B'</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Olive" title="b2">I2'</font> ex <font color="Olive" title="b4">f'</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I1</font>,<font color="Olive" title="b2">I2'</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f'</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B'</font> &amp; <font color="Olive" title="b1">IT'</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c4">A</font>,<font color="Olive" title="b3">B'</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Olive" title="b4">f'</font> )</span><br/><b>then reconsider </b><font color="Maroon" title="c6">I2'</font> = <font color="Maroon" title="c2">I2</font> as  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>  ;<br/><b>reconsider </b><font color="Maroon" title="c7">f'</font> = <font color="Maroon" title="c3">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I1</font>,<font color="Maroon" title="c6">I2'</font> ;<br/><b>reconsider </b><font color="Maroon" title="c8">B'</font> = <font color="Maroon" title="c5">B</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c6">I2'</font> ;<br/><b>consider </b><font color="Maroon" title="c9">IT</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">B'</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c7">f'</font>;<br/><b>reconsider </b><font color="Maroon" title="c10">IT'</font> = <font color="Maroon" title="c9">IT</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I1</font> ;<br/><b>take </b><font color="Maroon" title="c11">IT'</font> = <font color="Maroon" title="c10">IT'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">I2'</font> being 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>  ex <font color="Olive" title="b2">B'</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Olive" title="b1">I2'</font> ex <font color="Olive" title="b3">f'</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I1</font>,<font color="Olive" title="b1">I2'</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f'</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B'</font> &amp; <font color="Maroon" title="c11">IT'</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c4">A</font>,<font color="Olive" title="b2">B'</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Olive" title="b3">f'</font> )</span><br/><b>take </b><font color="Maroon" title="c12">I2'</font> = <font color="Maroon" title="c6">I2'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">B'</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c12">I2'</font> ex <font color="Olive" title="b2">f'</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I1</font>,<font color="Maroon" title="c12">I2'</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f'</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B'</font> &amp; <font color="Maroon" title="c11">IT'</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c4">A</font>,<font color="Olive" title="b1">B'</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Olive" title="b2">f'</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c13">f'</font> = <font color="Maroon" title="c3">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I1</font>,<font color="Maroon" title="c12">I2'</font> ;<br/><b>reconsider </b><font color="Maroon" title="c14">B'</font> = <font color="Maroon" title="c5">B</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c12">I2'</font> ;<br/><b>take </b><font color="Maroon" title="c15">B'</font> = <font color="Maroon" title="c14">B'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f'</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I1</font>,<font color="Maroon" title="c12">I2'</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f'</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">B'</font> &amp; <font color="Maroon" title="c11">IT'</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c15">B'</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Olive" title="b1">f'</font> )</span><br/><b>take </b><font color="Maroon" title="c16">f'</font> = <font color="Maroon" title="c13">f'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f'</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">B'</font> &amp; <font color="Maroon" title="c11">IT'</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c15">B'</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c16">f'</font> )</span><br/><b>thus </b><a NAME="E9:24_1_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f'</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">B'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">IT'</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c15">B'</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c16">f'</font></span><br/><b>thus </b><a NAME="E10:24_1_1_1"/>
<font color="Maroon" title="c11">IT'</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c15">B'</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c16">f'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:24_1_1"/>
( not <font color="Maroon" title="c2">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I1</font> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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