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<span class="kw">let </span><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> holds <br/>( <font color="Olive" title="b1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">S2</font> &amp; <font color="Olive" title="b1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">S2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> implies ( <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">S2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">S2</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:48"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">S2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">S2</font> )</span><br/>

<a NAME="E2:48"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="circtrm1.html#D9" target="_self" title="CIRCTRM1:def.9">Def9</a></span>;<br/>
<a NAME="E3:48"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="circtrm1.html#D9" target="_self" title="CIRCTRM1:def.9">Def9</a></span>;<br/>
<a NAME="E4:48"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="circtrm1.html#T25" target="_self" title="CIRCTRM1:th.25">Th25</a></span>;<br/>
<a NAME="E5:48"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E35">A6</font></span>: <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S1</font></span>)</span> = 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T160" title="RELAT_1:th.160">RELAT_1:160</a></span>
<br/>.= 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></span>
<br/>.= 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">S2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></span>
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">S1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="funct_1.html#T123" title="FUNCT_1:th.123">FUNCT_1:123</a></span>
<br/>.= 
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">S2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="circtrm1.html#T24" target="_self" title="CIRCTRM1:th.24">Th24</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">S2</font></span><br/>

<span class="kw">thus </span><a NAME="E8:48"/>
<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon" title="c2">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E35">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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