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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="pencil_1.html#V15" target="_self">PLS-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>thus </b><a NAME="E1:61"/>
( <font color="Maroon">c<sub>3</sub></font> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being non <a href="pencil_1.html#V13" target="_self">trivial-yielding</a> <a href="pencil_1.html#V16" target="_self">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="pencil_1.html#V16" target="_self">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:61_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D22" target="_self">Def22</a></i>;<br/>
<a NAME="E4:61_1"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E1:61_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pencil_1.html#E3:61_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pencil_1.html#D6" target="_self">Def6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="pencil_1.html#V13" target="_self">trivial-yielding</a> <a href="pencil_1.html#V16" target="_self">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="pencil_1.html#T13" target="_self">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:61_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_1.html#E3:61_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:61_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( not <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E3:61_1_1"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_1.html#E7:61_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="pencil_1.html#D6" target="_self">Def6</a></i>;<br/><b>hence </b><a NAME="E4:61_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_1.html#E2:61_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="pencil_1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:61_1"/>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="pencil_1.html#V13" target="_self">trivial-yielding</a> <a href="pencil_1.html#V16" target="_self">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="pencil_1.html#E3:61_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="pencil_1.html#E7:61_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>4</sub></font> being  non <a href="pencil_1.html#V13" target="_self">trivial-yielding</a> <a href="pencil_1.html#V16" target="_self">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E2:61"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2" target="_self">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" target="_self">indx</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 ;<br/>

<b>thus </b><a NAME="E3:61"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" target="_self">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_1.html#E2:61"><i><font color="Green">E41</font></i></a>, <a class="ref" href="pencil_1.html#D22" target="_self">Def22</a></i>;<br/>


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