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<span class="kw">let </span><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">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"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">G</font> implies ( <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  iff  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font></span>)</span> st <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">P</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:95"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">G</font>
 ; <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">F</font>,<font color="Maroon" title="c2">G</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  iff  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font></span>)</span> st <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">P</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:95"/>
( <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  implies  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font></span>)</span> st <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">P</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font></span>)</span> st <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">P</font> implies <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:95_1"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font></span>)</span> st <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">P</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">I</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:95_1"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">and </span><br/><a NAME="E4:95_1"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E5:95_1"/><span class="lab"><font color="Green" title="E66">A5</font></span>: 
<font color="Maroon" title="c3">I</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="E6:95_1"/><span class="lab"><font color="Green" title="E67">A6</font></span>: 
<font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="classes1.html#T85" target="_self" title="CLASSES1:th.85">Th85</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">I</font> = <font color="Maroon" title="c3">I</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E3:95_1"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="txt" href="#E4:95_1"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">I</font> = <font color="Maroon" title="c4">I</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E4:95_1"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="txt" href="#E5:95_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="ref" href="funct_2.html#T83" title="FUNCT_2:th.83">FUNCT_2:83</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">I</font>
; <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">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">I</font></span><br/>

<span class="kw">thus </span><a NAME="E9:95_1"/>
<font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95_1"><span class="lab"><font color="Green" title="E67">A6</font></span></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/>


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

<span class="kw">given </span><font color="Maroon" title="c3">P</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font></span>)</span><span class="kw"> such that </span><a NAME="E4:95"/><span class="lab"><font color="Green" title="E63">A7</font></span>: 
<font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">P</font>
 ; <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">F</font>,<font color="Maroon" title="c2">G</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> </span><br/>

<a NAME="E5:95"/><span class="lab"><font color="Green" title="E64">A8</font></span>: 
( <font color="Maroon" title="c3">P</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<a NAME="E6:95"/><span class="lab"><font color="Green" title="E65">A9</font></span>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:95"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:95"/>
<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E4:95"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="txt" href="#E6:95"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="ref" href="classes1.html#T85" target="_self" title="CLASSES1:th.85">Th85</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/>


</div>
