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<span class="kw">let </span><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</font> be   <a href="stacks_1.html#NM3" title="STACKS_1:NM.3">StackAlgebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font> <a href="stacks_1.html#R4" title="STACKS_1:pred.4">are_isomorphic</a>  &amp; <font color="Maroon" title="c1">X1</font> is  <a href="stacks_1.html#V7" title="STACKS_1:attr.7">proper-for-identity</a>  implies <font color="Maroon" title="c2">X2</font> is  <a href="stacks_1.html#V7" title="STACKS_1:attr.7">proper-for-identity</a>  )</span><br/>



<span class="kw">given </span><font color="Maroon" title="c3">F</font>, <font color="Maroon" title="c4">G</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E2:91"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">G</font> <a href="stacks_1.html#R3" title="STACKS_1:pred.3">form_isomorphism_between</a> <font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="stacks_1.html#D22" target="_self" title="STACKS_1:def.22">STACKS_1:def 22</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">X1</font> is  <a href="stacks_1.html#V7" title="STACKS_1:attr.7">proper-for-identity</a>  or <font color="Maroon" title="c2">X2</font> is  <a href="stacks_1.html#V7" title="STACKS_1:attr.7">proper-for-identity</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:91"/><span class="lab"><font color="Green" title="E62">A2</font></span>: 
<font color="Maroon" title="c1">X1</font> is  <a href="stacks_1.html#V7" title="STACKS_1:attr.7">proper-for-identity</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X2</font> is  <a href="stacks_1.html#V7" title="STACKS_1:attr.7">proper-for-identity</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</font> be   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c2">X2</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="stacks_1.html#D15" target="_self" title="STACKS_1:def.15">STACKS_1:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">s1</font> <a href="stacks_1.html#R2" title="STACKS_1:pred.2">==</a> <font color="Maroon" title="c6">s2</font> implies <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font> )</span><br/>

<a NAME="E4:91"/><span class="lab"><font color="Green" title="E63">A3</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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="c1">X1</font> &amp;  <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">X2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="stacks_1.html#D21" target="_self" title="STACKS_1:def.21">ISO</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">q1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:91"/><span class="lab"><font color="Green" title="E64">A4</font></span>: 
( <font color="Maroon" title="c7">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">q1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">q2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:91"/><span class="lab"><font color="Green" title="E65">A5</font></span>: 
( <font color="Maroon" title="c8">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">q2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">q1</font> = <font color="Maroon" title="c7">q1</font>, <font color="Maroon" title="c10">q2</font> = <font color="Maroon" title="c8">q2</font> as   <a href="stacks_1.html#NM2" title="STACKS_1:NM.2">stack</a> of <font color="Maroon" title="c1">X1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="stacks_1.html#D21" target="_self" title="STACKS_1:def.21">ISO</a>, <a class="txt" href="#E6:91"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="txt" href="#E8:91"><span class="lab"><font color="Green" title="E65">A5</font></span></a></span>;<br/>
<a NAME="E10:91"/><span class="lab"><font color="Green" title="E66">A6</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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">X1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</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="c2">X2</font> &amp; <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="#E2:91"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="stacks_1.html#D21" target="_self" title="STACKS_1:def.21">ISO</a></span>;<br/>
<a NAME="E11:91"/><span class="lab"><font color="Green" title="E67">A7</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c9">q1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X1</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">q2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X1</font> )
 
;<br/>
<span class="kw">assume </span><a NAME="E12:91"/>
<span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c5">s1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">s2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="stacks_1.html#D14" target="_self" title="STACKS_1:def.14">STACKS_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font></span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E68">A8</font></span>: <font color="Maroon" title="c3">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c9">q1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> = 
<span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">s2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E6:91"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="ref" href="stacks_1.html#T45" target="_self" title="STACKS_1:th.45">Th97</a></span>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">q2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E8:91"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="ref" href="stacks_1.html#T45" target="_self" title="STACKS_1:th.45">Th97</a></span>
;<br/>
<a NAME="E14:91"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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> <span class="p2"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c9">q1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c9">q1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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> <span class="p2"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">q2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">q2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:91"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="txt" href="#E11:91"><span class="lab"><font color="Green" title="E67">A7</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<a NAME="E15:91"/><span class="kw">then </span>
<span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c9">q1</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="stacks_1.html#K9" title="STACKS_1:func.9">|.</a><span class="default"><font color="Maroon" title="c10">q2</font></span><a href="stacks_1.html#K9" title="STACKS_1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:91"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="txt" href="#E11:91"><span class="lab"><font color="Green" title="E67">A7</font></span></a>, <a class="txt" href="#E13:91"><span class="lab"><font color="Green" title="E68">A8</font></span></a>, <a class="ref" href="funct_1.html#T27" title="FUNCT_1:th.27">FUNCT_1:27</a></span>;<br/>
<a NAME="E16:91"/><span class="kw">then </span>
<font color="Maroon" title="c9">q1</font> <a href="stacks_1.html#R2" title="STACKS_1:pred.2">==</a> <font color="Maroon" title="c10">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stacks_1.html#D14" target="_self" title="STACKS_1:def.14">CONG</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:91"/>
<font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="txt" href="#E6:91"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="txt" href="#E8:91"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="ref" href="stacks_1.html#D15" target="_self" title="STACKS_1:def.15">PROP</a></span>; <a class="txt" 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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