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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>3</sub></font>()] implies  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>3</sub></font>() <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Maroon" title="c1">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>() )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>3</sub></font>()]
 <span class="kw">and </span><br/><a NAME="E2:58_1"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c1">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>()
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow16.html#D6" target="_self" title="YELLOW16:def.6">YELLOW16:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span></span><br/>

<a NAME="E3:58_1"/>
 <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>3</sub></font>() is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="yellow16.html#T39" target="_self" title="YELLOW16:th.39">Th39</a></span>;<br/>
<a NAME="E4:58_1"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">Y</font> = <font color="Maroon" title="c1">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">f</font> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E44">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c4">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font></span>)</span></span><br/><a NAME="E1:58_1_1"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon">F<sub>2</sub></font>() <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="yellow16.html#T33" target="_self" title="YELLOW16:th.33">Th33</a></span>;<br/><a NAME="E2:58_1_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a>  <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon">F<sub>2</sub></font>() <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E1:58_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a></span>;<br/><a NAME="E3:58_1_1"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="yellow16.html#D6" target="_self" title="YELLOW16:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:58_1_1"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c4">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="yellow16.html#T35" target="_self" title="YELLOW16:th.35">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:58_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T27" title="WAYBEL_3:th.27">WAYBEL_3:27</a></span>;<br/>
<a NAME="E8:58_1"/><span class="kw">then </span>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="waybel_3.html#T27" title="WAYBEL_3:th.27">WAYBEL_3:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:58_1"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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