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<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</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">  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><br/>  for <font color="Olive" title="b3">C1</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b4">C2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b3">C1</font>,<font color="Olive" title="b4">C2</font> <a href="circtrm1.html#R5" title="CIRCTRM1:pred.5">are_similar_wrt</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> holds <br/> for <font color="Olive" title="b5">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b3">C1</font><br/>  for <font color="Olive" title="b6">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b4">C2</font>  st <font color="Olive" title="b5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font> holds <br/> for <font color="Olive" title="b7">v1</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( <font color="Olive" title="b5">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b7">v1</font> iff <font color="Olive" title="b6">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">v1</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" 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">  for <font color="Olive" title="b1">C1</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">C2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">C1</font>,<font color="Olive" title="b2">C2</font> <a href="circtrm1.html#R5" title="CIRCTRM1:pred.5">are_similar_wrt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> holds <br/> for <font color="Olive" title="b3">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b1">C1</font><br/>  for <font color="Olive" title="b4">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b2">C2</font>  st <font color="Olive" title="b3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b5">v1</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( <font color="Olive" title="b3">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b5">v1</font> iff <font color="Olive" title="b4">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">v1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">C1</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">C2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c5">C1</font>,<font color="Olive" title="b1">C2</font> <a href="circtrm1.html#R5" title="CIRCTRM1:pred.5">are_similar_wrt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> holds <br/> for <font color="Olive" title="b2">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font><br/>  for <font color="Olive" title="b3">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b1">C2</font>  st <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b4">v1</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( <font color="Olive" title="b2">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b4">v1</font> iff <font color="Olive" title="b3">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">v1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">C2</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</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="c5">C1</font>,<font color="Maroon" title="c6">C2</font> <a href="circtrm1.html#R5" title="CIRCTRM1:pred.5">are_similar_wrt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> implies  for <font color="Olive" title="b1">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font><br/>  for <font color="Olive" title="b2">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</font>  st <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b3">v1</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( <font color="Olive" title="b1">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b3">v1</font> iff <font color="Olive" title="b2">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">v1</font> ) )</span><br/>











<span class="kw">assume </span><a NAME="E1:75"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">C2</font> <a href="circtrm1.html#R5" title="CIRCTRM1:pred.5">are_similar_wrt</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</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">  for <font color="Olive" title="b1">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font><br/>  for <font color="Olive" title="b2">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</font>  st <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b3">v1</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( <font color="Olive" title="b1">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b3">v1</font> iff <font color="Olive" title="b2">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">v1</font> )</span><br/>

<a NAME="E2:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon" title="c6">C2</font>,<font color="Maroon" title="c5">C1</font> <a href="circtrm1.html#R5" title="CIRCTRM1:pred.5">are_similar_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 class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#T39" target="_self" onmouseover="rs('circtrm1/T39')" onmouseout="rh()">Th39</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">s1</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</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">  for <font color="Olive" title="b1">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</font>  st <font color="Maroon" title="c7">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b2">v1</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( <font color="Maroon" title="c7">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b2">v1</font> iff <font color="Olive" title="b1">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">s2</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</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="c7">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> implies  for <font color="Olive" title="b1">v1</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( <font color="Maroon" title="c7">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b1">v1</font> iff <font color="Maroon" title="c8">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E3:75"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c7">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</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">  for <font color="Olive" title="b1">v1</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( <font color="Maroon" title="c7">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b1">v1</font> iff <font color="Maroon" title="c8">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">v1</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">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="c7">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c9">v1</font> iff <font color="Maroon" title="c8">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font> )</span><br/>

<a NAME="E4:75"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="circtrm1.html#T37" target="_self" onmouseover="rs('circtrm1/T37')" onmouseout="rh()">Th37</a></span>;<br/>
<a NAME="E5:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A5</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">G1</font>,<font color="Maroon" title="c2">G2</font>
 
;<br/>
<a NAME="E6:75"/><span class="lab"><font color="Green" title="E56">A6</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">G2</font>,<font color="Maroon" title="c1">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75"><span class="lab"><font color="Green" title="E54">A4</font></span></a></span>;<br/>
<a NAME="E7:75"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E55">A5</font></span></a></span>;<br/>
<a NAME="E8:75"/><span class="lab"><font color="Green" title="E58">A8</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75"><span class="lab"><font color="Green" title="E56">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">v2</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font> as   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c2">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="circtrm1.html#T30" target="_self" onmouseover="rs('circtrm1/T30')" onmouseout="rh()">Th30</a></span>;<br/>
<a NAME="E10:75"/>
<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="#E4:75"><span class="lab"><font color="Green" title="E54">A4</font></span></a></span>;<br/>
<a NAME="E11:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="funct_1.html#T34" onmouseover="rs('funct_1/T34')" onmouseout="rh()">FUNCT_1:34</a></span>;<br/>
<span class="kw">thus </span><a NAME="E12:75"/>
( <font color="Maroon" title="c7">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c8">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</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="c8">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c7">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c9">v1</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="75_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:75_1"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s1</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="facirc_1.html#D9" onmouseover="rs('facirc_1/D9')" onmouseout="rh()">FACIRC_1:def 9</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="c8">s2</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="facirc_1.html#D9" onmouseover="rs('facirc_1/D9')" onmouseout="rh()">FACIRC_1:def 9</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"> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c11">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span>)</span></span><br/>

<a NAME="E2:75_1"/>
<font color="Maroon" title="c11">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:75_1"/><span class="kw">then </span>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c11">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c11">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="circtrm1.html#T55" target="_self" onmouseover="rs('circtrm1/T55')" onmouseout="rh()">Th55</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c11">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c11">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1"><span class="lab"><font color="Green" title="E60">A10</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
;<br/> <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/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E13:75"/><span class="lab"><font color="Green" title="E60">A11</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s2</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span>)</span>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="facirc_1.html#D9" onmouseover="rs('facirc_1/D9')" onmouseout="rh()">FACIRC_1:def 9</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="c7">s1</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c9">v1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="facirc_1.html#D9" onmouseover="rs('facirc_1/D9')" onmouseout="rh()">FACIRC_1:def 9</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"> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c11">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font></span><br/>

<a NAME="E14:75"/><span class="lab"><font color="Green" title="E61">A12</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E15:75"/>
<font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="circtrm1.html#T51" target="_self" onmouseover="rs('circtrm1/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<a NAME="E16:75"/><span class="kw">then </span>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c11">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c11">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E14:75"><span class="lab"><font color="Green" title="E61">A12</font></span></a>, <a class="ref" href="circtrm1.html#T55" target="_self" onmouseover="rs('circtrm1/T55')" onmouseout="rh()">Th55</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c11">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font> = 
<span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c11">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:75"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E11:75"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c8">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:75"><span class="lab"><font color="Green" title="E60">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
;<br/> <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/>


</div>
