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<b>reconsider </b><font color="Maroon" title="c5">mF</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c2">o</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E2:6_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">F</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c5">mF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</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>consider </b><font color="Maroon" title="c6">mo</font> being   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:6_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">mo</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> ) ) )
 <b>by </b><i><a class="txt" href="ortsp_1.html#E5"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">MPS</font> =  <a href="symsp_1.html#G1" title="SYMSP_1:aggr.1">SymStr</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">md</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">mF</font>,<font color="Maroon" title="c6">mo</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="symsp_1.html#L1" title="SYMSP_1:struct.1">SymStr</a> of <font color="Maroon" title="c4">F</font> <b>by </b><i><a class="txt" href="ortsp_1.html#E6"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">MPS</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">MPS</font> is <a href="ortsp_1.html#V1" title="ORTSP_1:attr.1">OrtSp-like</a> &amp; <font color="Maroon" title="c7">MPS</font> is <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">VectSp-like</a> &amp; <font color="Maroon" title="c7">MPS</font> is <a href="symsp_1.html#V1" title="SYMSP_1:attr.1">strict</a> )</span><br/>

<b>thus </b><a NAME="E6:6_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">MPS</font><br/> for <font color="Olive" title="b6">l</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">F</font> holds <br/> ( ( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c7">MPS</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c7">MPS</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c7">MPS</font> &amp; <font color="Olive" title="b4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c7">MPS</font> implies  ex <font color="Olive" title="b7">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">MPS</font> st <br/>( not <font color="Olive" title="b1">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Olive" title="b2">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Olive" title="b3">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Olive" title="b4">d</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) ) &amp; ( <font color="Olive" title="b2">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  implies <font color="Olive" title="b2">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) &amp; ( <font color="Olive" title="b1">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Olive" title="b1">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  implies <font color="Olive" title="b1">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) &amp; ( not <font color="Olive" title="b1">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  implies  ex <font color="Olive" title="b7">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">F</font> st <font color="Olive" title="b1">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) &amp; ( <font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b3">c</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; <font color="Olive" title="b3">c</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b1">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  implies <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) )
 <b>by </b><i><a class="txt" href="ortsp_1.html#E2:6_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="ortsp_1.html#E4:6_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="ortsp_1.html#E8"><i><font color="Green" title="E5">Lm5</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ortsp_1.html#D2" target="_self" title="ORTSP_1:def.2">ORTSP_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">MPS</font> is <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">VectSp-like</a> &amp; <font color="Maroon" title="c7">MPS</font> is <a href="symsp_1.html#V1" title="SYMSP_1:attr.1">strict</a> )</span><br/>

<b>thus </b><a NAME="E7:6_1_1"/>
<font color="Maroon" title="c7">MPS</font> is <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">VectSp-like</a>
 <b>by </b><i><a class="txt" href="ortsp_1.html#E2:6_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="ortsp_1.html#E7"><i><font color="Green" title="E4">Lm4</font></i></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="c7">MPS</font> is <a href="symsp_1.html#V1" title="SYMSP_1:attr.1">strict</a></span><br/>

<b>thus </b><a NAME="E8:6_1_1"/>
<font color="Maroon" title="c7">MPS</font> is <a href="symsp_1.html#V1" title="SYMSP_1:attr.1">strict</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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