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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" 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="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="circtrm1.html#R2" title="CIRCTRM1:pred.2">are_equivalent</a>  &amp; <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">S3</font> <a href="circtrm1.html#R2" title="CIRCTRM1:pred.2">are_equivalent</a>  implies <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">S3</font> <a href="circtrm1.html#R2" title="CIRCTRM1:pred.2">are_equivalent</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">g1</font> being   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E2:49"/><span class="lab"><font color="Green" title="E29">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="c4">f1</font>,<font color="Maroon" title="c5">g1</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="circtrm1.html#D10" target="_self" onmouseover="rs('circtrm1/D10')" onmouseout="rh()">CIRCTRM1:def 10</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"> (  not <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">S3</font> <a href="circtrm1.html#R2" title="CIRCTRM1:pred.2">are_equivalent</a>  or <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">S3</font> <a href="circtrm1.html#R2" title="CIRCTRM1:pred.2">are_equivalent</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c6">f2</font>, <font color="Maroon" title="c7">g2</font> being   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E4:49"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<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>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="circtrm1.html#D10" target="_self" onmouseover="rs('circtrm1/D10')" onmouseout="rh()">CIRCTRM1:def 10</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="c1">S1</font>,<font color="Maroon" title="c3">S3</font> <a href="circtrm1.html#R2" title="CIRCTRM1:pred.2">are_equivalent</a> </span><br/>

<span class="kw">take </span>
<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 class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="circtrm1.html#D10" target="_self" onmouseover="rs('circtrm1/D10')" onmouseout="rh()">CIRCTRM1:def 10</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">  ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <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="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="Olive" title="b1">g</font></span><br/>

<span class="kw">take </span>
<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 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="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="E5:49"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="circtrm1.html#T27" target="_self" onmouseover="rs('circtrm1/T27')" onmouseout="rh()">Th27</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"> verum</span><br/>


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