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<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">F</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = <font color="Maroon" title="c2">o</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">mF</font> being   <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>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><span class="kw"> such that </span><br/><a NAME="E2:11_1_1"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 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 <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds  <font color="Maroon" title="c5">mF</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S4" onmouseover="rs('binop_1/S4')" onmouseout="rh()">BINOP_1:sch 4</a>();<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c6">mo</font> being   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">Relation</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><span class="kw"> such that </span><br/><a NAME="E4:11_1_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">mo</font> iff ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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 <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> 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> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="symsp_1.html#E4"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">MPS</font> =  <a href="symsp_1.html#G1" title="SYMSP_1:aggr.1">SymStr</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<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> over <font color="Maroon" title="c4">F</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="symsp_1.html#E5"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">MPS</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">MPS</font> is  <a href="symsp_1.html#V2" title="SYMSP_1:attr.2">SymSp-like</a>  &amp; <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp; <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp; <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp; <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</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/>

<span class="kw">thus </span><a NAME="E6:11_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">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="b5">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> implies  ex <font color="Olive" title="b6">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">MPS</font> st  not <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="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="b6">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#K3" title="RLVECT_1:func.3">+</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#K3" title="RLVECT_1:func.3">+</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#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">b</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="symsp_1.html#E7"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="symsp_1.html#D1" target="_self" onmouseover="rs('symsp_1/D1')" onmouseout="rh()">SYMSP_1:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp; <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp; <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp; <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</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/>

<span class="kw">thus </span><a NAME="E7:11_1_1"/>
( <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp; <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp; <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp; <font color="Maroon" title="c7">MPS</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="symsp_1.html#E6"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E8:11_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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