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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>





<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow16.html#D1" target="_self" title="YELLOW16:def.1">Def1</a></span>;<br/>
<a NAME="E3:20"/>
<font color="Maroon" title="c3">f</font> is  <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="yellow16.html#T14" target="_self" title="YELLOW16:th.14">Th14</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E16">A2</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quantal1.html#D9" title="QUANTAL1:def.9">QUANTAL1:def 9</a></span>
<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#K2" title="RELSET_1:func.2">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T2" title="FUNCT_4:th.2">FUNCT_4:2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></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></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="yellow16.html#D1" target="_self" title="YELLOW16:def.1">Def1</a></span>
<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="E5:20"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="yellow16.html#D1" target="_self" title="YELLOW16:def.1">Def1</a></span>;<br/>
<a NAME="E6:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A4</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel21.html#T10" title="WAYBEL21:th.10">WAYBEL21:10</a></span>;<br/>
<a NAME="E7:20"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></span>;<br/>
<a NAME="E8:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A5</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D1" title="YELLOW_9:def.1">YELLOW_9:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:20"/>
<font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="waybel20.html#T29" title="WAYBEL20:th.29">WAYBEL20:29</a></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" title="c3">f</font> is  <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a> </span><br/>

<a NAME="E10:20"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E19">A5</font></span></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></span>;<br/>
<span class="kw">hence </span><a NAME="E11:20"/>
<font color="Maroon" title="c3">f</font> is  <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#D13" title="WAYBEL_1:def.13">WAYBEL_1:def 13</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/>


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