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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 <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="pencil_1.html#V11" title="PENCIL_1:attr.11">TopStruct-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( not <font color="Olive" title="b1">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b2">i</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> &amp; not for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Olive" title="b1">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b3">i</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> ) ) holds <br/> not  <a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Olive" title="b1">A</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="pencil_1.html#V11" title="PENCIL_1:attr.11">TopStruct-yielding</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">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( not <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> &amp; not for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b2">i</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> ) ) implies  not  <a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> )</span><br/>



<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E35">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( not <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a> &amp; not for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b2">i</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not  <a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a></span><br/>

<a NAME="E2:52"/><b>then </b>
not  <a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T15" target="_self" title="PENCIL_1:th.15">Th15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">SB</font> =  <a href="pencil_1.html#K4" title="PENCIL_1:func.4">Segre_Blocks</a> <font color="Maroon" title="c2">A</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">SB</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c4">B</font> being  <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green" title="E36">A2</font></i>: 
(  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c4">B</font> &amp;  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> st <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D22" target="_self" title="PENCIL_1:def.22">Def22</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E5:52_1"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:52_1"><i><font color="Green" title="E36">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">R</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E7:52_1"/><i><font color="Green" title="E38">A4</font></i>: 
( <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )
 <b>by </b><i><a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><a NAME="E8:52_1"/>
<font color="Maroon" title="c4">B</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 <b>by </b><i><a class="txt" href="#E3:52_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="pencil_1.html#T1" target="_self" title="PENCIL_1:th.1">Th1</a></i>;<br/><a NAME="E9:52_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:52_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E5:52_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="pua2mss1.html#T2" title="PUA2MSS1:th.2">PUA2MSS1:2</a></i>;<br/><a NAME="E10:52_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K1" title="PENCIL_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a>
 <b>by </b><i><a class="txt" href="#E7:52_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="pencil_1.html#D5" target="_self" title="PENCIL_1:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E11:52_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:52"><i><font color="Green" title="E35">A1</font></i></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>
<a NAME="E5:52"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> is  not    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">SB</font>
 
;<br/>
<b>hence </b><a NAME="E6:52"/>
not  <a href="pencil_1.html#K5" title="PENCIL_1:func.5">Segre_Product</a> <font color="Maroon" title="c2">A</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D5" target="_self" title="PENCIL_1:def.5">Def5</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>
