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<b>thus </b><a NAME="E1:97_1_1"/>
( <font color="Maroon" title="c1">T</font> is <a href="abcmiz_0.html#V12" title="ABCMIZ_0:attr.12">non-absorbing</a> implies for <font color="Olive" title="b1">a</font> being  <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font> holds   <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Olive" title="b1">a</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">a</font> being  <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font> holds   <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Olive" title="b1">a</font> ) implies <font color="Maroon" title="c1">T</font> is <a href="abcmiz_0.html#V12" title="ABCMIZ_0:attr.12">non-absorbing</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_1_1_1"/><i><font color="Green" title="E56">A1</font></i>: 
the <a href="abcmiz_0.html#U4" title="ABCMIZ_0:sel.4">sub-map</a> of <font color="Maroon" title="c1">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c1">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="abcmiz_0.html#U4" title="ABCMIZ_0:sel.4">sub-map</a> of <font color="Maroon" title="c1">T</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D24" target="_self" title="ABCMIZ_0:def.24">ABCMIZ_0:def 24</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">a</font> being  <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font> holds   <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Olive" title="b1">a</font></span><br/>

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <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">  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Maroon" title="c2">a</font></span><br/>

<b>thus </b><a NAME="E2:97_1_1_1"/>
 <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="#E1:97_1_1_1"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</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>

<b>assume </b><a NAME="E2:97_1_1"/><i><font color="Green" title="E56">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font> holds   <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Olive" title="b1">a</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">T</font> is <a href="abcmiz_0.html#V12" title="ABCMIZ_0:attr.12">non-absorbing</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <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"> <span class="p1">(<span class="default">the <a href="abcmiz_0.html#U4" title="ABCMIZ_0:sel.4">sub-map</a> of <font color="Maroon" title="c1">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="abcmiz_0.html#U4" title="ABCMIZ_0:sel.4">sub-map</a> of <font color="Maroon" title="c1">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span><br/><b>reconsider </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">x</font> as   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font> ;<br/><b>thus </b><span class="p1">(<span class="default">the <a href="abcmiz_0.html#U4" title="ABCMIZ_0:sel.4">sub-map</a> of <font color="Maroon" title="c1">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> = 
 <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c3">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="#E2:97_1_1"><i><font color="Green" title="E56">A2</font></i></a></i>
<br/>.= 
the <a href="abcmiz_0.html#U4" title="ABCMIZ_0:sel.4">sub-map</a> of <font color="Maroon" title="c1">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</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>
<b>hence </b><a NAME="E4:97_1_1"/>
the <a href="abcmiz_0.html#U4" title="ABCMIZ_0:sel.4">sub-map</a> of <font color="Maroon" title="c1">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c1">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="abcmiz_0.html#U4" title="ABCMIZ_0:sel.4">sub-map</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D24" target="_self" title="ABCMIZ_0:def.24">ABCMIZ_0:def 24</a> <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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