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<b>let </b><font color="Maroon" title="c1">L</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">L</font>,<font color="Maroon" title="c1">L</font>  st  ex <font color="Olive" title="b2">T</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> ex <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b2">T</font> ex <font color="Olive" title="b4">i</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">T</font>,<font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">q</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Olive" title="b4">i</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Olive" title="b1">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">i</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">q</font> &amp;  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Olive" title="b2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">i</font> ) holds <br/><font color="Olive" title="b1">f</font> <a href="waybel_1.html#NR10" title="WAYBEL_1:NR.10">is_a_projection_operator</a> </span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">T</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> ex <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">T</font> ex <font color="Olive" title="b3">i</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">q</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Olive" title="b3">i</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">i</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">q</font> &amp;  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Olive" title="b1">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">i</font> ) implies <font color="Maroon" title="c2">f</font> <a href="waybel_1.html#NR10" title="WAYBEL_1:NR.10">is_a_projection_operator</a>  )</span><br/>



<b>given </b><font color="Maroon" title="c3">T</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>, <font color="Maroon" title="c4">q</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c3">T</font>, <font color="Maroon" title="c5">i</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c1">L</font><b> such that </b><a NAME="E2:76"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c4">q</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>and </b><br/><a NAME="E3:76"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c5">i</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>and </b><br/><a NAME="E4:76"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">i</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">q</font>
 <b>and </b><br/><a NAME="E5:76"/><i><font color="Green" title="E50">A4</font></i>: 
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">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="c2">f</font> <a href="waybel_1.html#NR10" title="WAYBEL_1:NR.10">is_a_projection_operator</a> </span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c5">i</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <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="c3">T</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E5:76"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E7:76"/>
<font color="Maroon" title="c2">f</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E4:76"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="waybel_1.html#T22" target="_self" title="WAYBEL_1:th.22">Th22</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D13" target="_self" title="WAYBEL_1:def.13">WAYBEL_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a></span><br/>

<b>thus </b><a NAME="E8:76"/>
<font color="Maroon" title="c2">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:76"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="waybel_1.html#E3:76"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="waybel_1.html#E4:76"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="yellow_2.html#T14" title="YELLOW_2:th.14">YELLOW_2:14</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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