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<b>let </b><font color="Maroon" title="c1">G</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</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">H1</font>, <font color="Olive" title="b2">H2</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of <font color="Maroon" title="c1">G</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</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="Olive" title="b2">H2</font> holds <br/><font color="Olive" title="b1">H1</font> is    <a href="monoid_0.html#M5" title="MONOID_0:mode.5">SubStr</a> of <font color="Olive" title="b2">H2</font></span><br/><b>let </b><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H1</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="c3">H2</font> implies <font color="Maroon" title="c2">H1</font> is    <a href="monoid_0.html#M5" title="MONOID_0:mode.5">SubStr</a> of <font color="Maroon" title="c3">H2</font> )</span><br/>



<b>assume </b><a NAME="E1:76"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">H2</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">H1</font> is    <a href="monoid_0.html#M5" title="MONOID_0:mode.5">SubStr</a> of <font color="Maroon" title="c3">H2</font></span><br/>

<a NAME="E2:76"/><b>then </b>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">H1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">H2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">H1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H1</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H1</font>)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H1</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H1</font>)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">H2</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">H2</font>)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">H2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">H2</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">H2</font>)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="monoid_0.html#D23" target="_self" title="MONOID_0:def.23">Def23</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E3:76"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">H2</font>)</span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H1</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">H1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H1</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">H2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">H2</font>) )
 
<b>by </b><i><a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a>, <a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a></i>;<br/>
<b>hence </b><a NAME="E4:76"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">H1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">H2</font>)
 <b>by </b><i><a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D23" target="_self" title="MONOID_0:def.23">MONOID_0:def 23</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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