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<b>reconsider </b><font color="Maroon" title="c1">K</font> = <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon">F<sub>3</sub></font>(), <font color="Maroon" title="c2">J</font> = <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon">F<sub>2</sub></font>() as  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="yellow16.html#V1" title="YELLOW16:attr.1">Poset-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<div><i><font color="Green" title="E41">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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" title="c1">K</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">K</font>] implies 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">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[ <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c3">X</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Olive" title="b1">i</font>] )</span><br/><b>assume </b><a NAME="E1:57_1_1"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">K</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">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[ <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c3">X</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Olive" title="b1">i</font>]</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font>]</span><br/><a NAME="E2:57_1_1"/>
<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:57_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font>]
 <b>by </b><i><a class="txt" href="#E1:57"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E1:57_1_1"><i><font color="Green" title="E42">A4</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>
<div><i><font color="Green" title="E42">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font></span><br/><a NAME="E1:57_1_2"/>
( <font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E2:57_1_2"/>
<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font>
 ; <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>
<div><i><font color="Green" title="E43">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font></span>)</span>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font>] holds <br/><font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font></span><br/><b>let </b><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font>] implies <font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> )</span><br/><b>assume </b><a NAME="E1:57_1_3"/><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font></span><br/><a NAME="E2:57_1_3"/>
( <font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:57_1_3"/>
<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="#E2:57"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E1:57_1_3"><i><font color="Green" title="E44">A7</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="E5:57_1"/>
for <font color="Olive" title="b1">X</font> being  <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" title="c1">K</font></span>)</span>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">K</font>] holds <br/> <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">K</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Olive" title="b1">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font>
 
<b>from </b><i><a class="ref" href="yellow16.html#S1" target="_self" title="YELLOW16:sch.1">YELLOW16:sch 1</a>(<a class="txt" href="#E2:57_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E3:57_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="#E4:57_1"><i><font color="Green" title="E43">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:57_1"/>
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon">F<sub>1</sub></font>()] holds <br/><font color="Maroon">F<sub>3</sub></font>() <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon">F<sub>1</sub></font>() <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon">F<sub>2</sub></font>() <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon">F<sub>1</sub></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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