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<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="waybel18.html#K2" target="_self">product_prebasis</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="pre_topc.html#V1">strict</a>  <a href="pre_topc.html#L1">TopStruct</a> <b> such that </b><br/><a NAME="E3:10_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> #)
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i><a class="txt" href="waybel18.html#E3:10_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="waybel18.html#T4" target="_self">Th4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E5:10_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:10_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:10_1_1_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:10_1_1_1"/><i><font color="Green">E8</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E6:10_1_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="waybel18.html#E3:10_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><a NAME="E7:10_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:10_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel18.html#E4:10_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="waybel18.html#E6:10_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a>  <b>by </b><i><a class="ref" href="waybel_3.html#D7">WAYBEL_3:def 7</a></i>;<br/><a NAME="E9:10_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="waybel18.html#E6:10_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/><b>hence </b><a NAME="E10:10_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E7:10_1_1"/><b>then </b>
not  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="card_3.html#T37">CARD_3:37</a></i>;<br/>
<b>hence </b><a NAME="E8:10_1_1"/>
 <a href="waybel18.html#K2" target="_self">product_prebasis</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:10_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cantor_1.html#T20">CANTOR_1:20</a></i>;<br/>


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