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<span class="kw">let </span><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c3">S3</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">f1</font>, <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">f2</font>, <font color="Olive" title="b4">g2</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">f1</font>,<font color="Olive" title="b2">g1</font> &amp; <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">S3</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Olive" title="b3">f2</font>,<font color="Olive" title="b4">g2</font> holds <br/><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">S3</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Olive" title="b3">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b4">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">g1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">g1</font>, <font color="Maroon" title="c6">f2</font>, <font color="Maroon" title="c7">g2</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="c4">f1</font>,<font color="Maroon" title="c5">g1</font> &amp; <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">S3</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c7">g2</font> implies <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">S3</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c4">f1</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="E2:47"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c5">g1</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="E3:47"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">g1</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="E4:47"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<font color="Maroon" title="c5">g1</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="kw">and </span><br/><a NAME="E5:47"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c6">f2</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="E6:47"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c7">g2</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="E7:47"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c7">g2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">S3</font>
 <span class="kw">and </span><br/><a NAME="E8:47"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<font color="Maroon" title="c7">g2</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="c3">S3</font>,<font color="Maroon" title="c2">S2</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="c1">S1</font>,<font color="Maroon" title="c3">S3</font> <a href="circtrm1.html#R1" title="CIRCTRM1:pred.1">are_equivalent_wrt</a> <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</font></span><br/>

<span class="kw">thus </span><a NAME="E9:47"/>
( <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</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="#E1:47"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E34">A6</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="c6">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</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="c3">S3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</font></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="c3">S3</font>,<font color="Maroon" title="c1">S1</font> )</span><br/>

<a NAME="E10:47"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="funct_1.html#T66" title="FUNCT_1:th.66">FUNCT_1:66</a></span>;<br/>
<a NAME="E11:47"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="funct_1.html#T66" title="FUNCT_1:th.66">FUNCT_1:66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:47"/>
( <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</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="c3">S3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</font></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="c3">S3</font>,<font color="Maroon" title="c1">S1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E8:47"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E10:47"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="pua2mss1.html#T30" title="PUA2MSS1:th.30">PUA2MSS1:30</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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