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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="Maroon" title="c2">S2</font>,<font color="Maroon" title="c1">S1</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<font color="Olive" title="b2">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </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="c2">S2</font>,<font color="Maroon" title="c1">S1</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46"/><span class="lab"><font color="Green" title="E28">A1</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">and </span><br/><a NAME="E2:46"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c4">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E3:46"/><span class="lab"><font color="Green" title="E30">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">and </span><br/><a NAME="E4:46"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c1">S1</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="circtrm1.html#D9" target="_self" title="CIRCTRM1:def.9">CIRCTRM1:def 9</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="c2">S2</font>,<font color="Maroon" title="c1">S1</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span><br/>

<span class="kw">thus </span><a NAME="E5:46"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  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="#E1:46"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="circtrm1.html#D9" target="_self" title="CIRCTRM1:def.9">CIRCTRM1:def 9</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="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c1">S1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <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><br/>

<a NAME="E6:46"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="funct_1.html#T65" title="FUNCT_1:th.65">FUNCT_1:65</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:46"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c1">S1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <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="#E2:46"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="funct_1.html#T65" title="FUNCT_1:th.65">FUNCT_1:65</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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