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<b>let </b><font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c4">T2</font> be  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">J</font></span>)</span> &amp;  <a href="waybel18.html#K2" title="WAYBEL18:func.2">product_prebasis</a> <font color="Maroon" title="c2">J</font> is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c3">T1</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">J</font></span>)</span> &amp;  <a href="waybel18.html#K2" title="WAYBEL18:func.2">product_prebasis</a> <font color="Maroon" title="c2">J</font> is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c4">T2</font> implies <font color="Maroon" title="c3">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:10_1_2"/><i><font color="Green" title="E6">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">J</font></span>)</span>
 <b>and </b><br/><a NAME="E2:10_1_2"/><i><font color="Green" title="E7">A6</font></i>: 
 <a href="waybel18.html#K2" title="WAYBEL18:func.2">product_prebasis</a> <font color="Maroon" title="c2">J</font> is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c3">T1</font>
 <b>and </b><br/><a NAME="E3:10_1_2"/><i><font color="Green" title="E8">A7</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">J</font></span>)</span>
 <b>and </b><br/><a NAME="E4:10_1_2"/><i><font color="Green" title="E9">A8</font></i>: 
 <a href="waybel18.html#K2" title="WAYBEL18:func.2">product_prebasis</a> <font color="Maroon" title="c2">J</font> is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c4">T2</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">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_2_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">J</font></span>)</span>
 ; <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="c5">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:10_1_2_1"/><i><font color="Green" title="E10">A9</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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">J</font></span>)</span> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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">J</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="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:10_1_2_1"/><i><font color="Green" title="E11">A10</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">J</font> <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>then 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="E6:10_1_2_1"/><i><font color="Green" title="E12">A11</font></i>: 
( <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R</font> )
 <b>by </b><i><a class="txt" href="#E3:10_1_2_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><a NAME="E7:10_1_2_1"/>
<font color="Maroon" title="c6">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">J</font>
 <b>by </b><i><a class="txt" href="#E3:10_1_2_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="#E4:10_1_2_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="#E6:10_1_2_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">R</font> = <font color="Maroon" title="c6">R</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  <b>by </b><i><a class="ref" href="waybel_3.html#D7" title="WAYBEL_3:def.7">WAYBEL_3:def 7</a></i>;<br/><a NAME="E9:10_1_2_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E6:10_1_2_1"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E10:10_1_2_1"/>
contradiction
 ; <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="E6:10_1_2"/><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">J</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_3.html#T37" title="CARD_3:th.37">CARD_3:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">t1</font> = <font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c6">t2</font> = <font color="Maroon" title="c4">T2</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> <b>by </b><i><a class="txt" href="#E1:10_1_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="#E3:10_1_2"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/>
<a NAME="E8:10_1_2"/>
<font color="Maroon" title="c5">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t2</font>
 
<b>by </b><i><a class="txt" href="#E1:10_1_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="#E2:10_1_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="#E3:10_1_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="#E4:10_1_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="cantor_1.html#T19" title="CANTOR_1:th.19">CANTOR_1:19</a></i>;<br/>
<b>hence </b><a NAME="E9:10_1_2"/>
<font color="Maroon" title="c3">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T2</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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