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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <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">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a><br/> for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">f</font> <a href="yellow16.html#R1" title="YELLOW16:pred.1">is_a_retraction_of</a> <font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">S</font> holds <br/>( <font color="Olive" title="b2">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> &amp; <font color="Olive" title="b2">f</font> is <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a> )</span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <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">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">f</font> <a href="yellow16.html#R1" title="YELLOW16:pred.1">is_a_retraction_of</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</font> holds <br/>( <font color="Olive" title="b1">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> &amp; <font color="Olive" title="b1">f</font> is <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a> )</span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</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">f</font> <a href="yellow16.html#R1" title="YELLOW16:pred.1">is_a_retraction_of</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</font> implies ( <font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a> ) )</span><br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="yellow16.html#R1" title="YELLOW16:pred.1">is_a_retraction_of</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</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">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">f</font> as  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="yellow16.html#D1" target="_self" title="YELLOW16:def.1">Def1</a></i>;<br/>
<a NAME="E3:18"/>
<font color="Maroon" title="c3">f</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a>
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="yellow16.html#T14" target="_self" title="YELLOW16:th.14">Th14</a></i>;<br/>
<b>then </b><i><font color="Green" title="E16">A2</font></i>: <font color="Maroon" title="c3">f</font> = 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="ref" href="quantal1.html#D9" title="QUANTAL1:def.9">QUANTAL1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="ref" href="funct_4.html#T2" title="FUNCT_4:th.2">FUNCT_4:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="yellow16.html#T11" target="_self" title="YELLOW16:th.11">Th11</a>, <a class="ref" href="yellow16.html#T13" target="_self" title="YELLOW16:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="yellow16.html#D1" target="_self" title="YELLOW16:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">g</font>

;<br/>
<a NAME="E5:18"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c2">S</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="yellow16.html#D1" target="_self" title="YELLOW16:def.1">Def1</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green" title="E18">A4</font></i>: 
 <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel21.html#T10" title="WAYBEL21:th.10">WAYBEL21:10</a></i>;<br/>
<a NAME="E7:18"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <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 <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="#E5:18"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E8:18"/><b>then </b><i><font color="Green" title="E19">A5</font></i>: 
 <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#D1" title="YELLOW_9:def.1">YELLOW_9:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:18"/>
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 <b>by </b><i><a class="txt" href="#E4:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E6:18"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="waybel20.html#T29" title="WAYBEL20:th.29">WAYBEL20:29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> is <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a></span><br/>

<a NAME="E10:18"/>
<font color="Maroon" title="c3">f</font> is  <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a> <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E4:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E6:18"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E8:18"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="yellow16.html#T14" target="_self" title="YELLOW16:th.14">Th14</a>, <a class="ref" href="waybel20.html#T29" title="WAYBEL20:th.29">WAYBEL20:29</a></i>;<br/>
<b>hence </b><a NAME="E11:18"/>
<font color="Maroon" title="c3">f</font> is <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D13" title="WAYBEL_1:def.13">WAYBEL_1:def 13</a></i>; <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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