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<b>let </b><font color="Maroon" title="c1">I</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">N</font> being non <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b1">A</font></span>)</span>,<font color="Olive" title="b2">N</font> is <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a> &amp; not <font color="Olive" title="b1">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b1">A</font></span>)</span>,<font color="Olive" title="b2">N</font> is <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being non <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Olive" title="b1">N</font> is <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a> &amp; not <font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Olive" title="b1">N</font> is <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> )</span><br/><b>let </b><font color="Maroon" title="c3">N</font> be  non <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font> is <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a> &amp; not <font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font> is <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> )</span><br/>





<b>thus </b><a NAME="E1:64"/>
<font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font> is <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>
  <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">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font> is <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D20" target="_self" title="PENCIL_1:def.20">PENCIL_1:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>  st  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">j</font> holds <br/>( not <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )</span><br/>

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font> implies ( not <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> ) )</span><br/>

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )</span><br/>

<a NAME="E2:64_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1"/>
( not <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="#E1:64_1"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="pencil_1.html#T12" target="_self" title="PENCIL_1:th.12">Th12</a></i>; <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="E2:64"/>
not <font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font> is <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D16" target="_self" title="PENCIL_1:def.16">PENCIL_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )</span><br/>

<a NAME="E1:64_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:64_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a></span><br/>

<a NAME="E3:64_2"/>
<font color="Maroon" title="c1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:64_2"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c2">A</font></span>)</span> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>; <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>


</div>
