<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">U1</font>, <font color="Maroon" title="c3">U2</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c4">H</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font>;<br/><b>let </b><font color="Maroon" title="c5">H1</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font>;<br/>







<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c4">H</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c5">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">H</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  )
 ;<br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c4">H</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">H</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#D12" target="_self" title="MSUALG_3:def.12">Def12</a></i>;<br/>
<a NAME="E3:31"/><b>then </b><i><font color="Green" title="E28">A3</font></i>: 
( <font color="Maroon" title="c4">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">H</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#D11" target="_self" title="MSUALG_3:def.11">Def11</a></i>;<br/>
<a NAME="E4:31"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c4">H</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E2:31"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="msualg_3.html#D10" target="_self" title="MSUALG_3:def.10">Def10</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green" title="E30">A5</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">H1</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Olive" title="b2">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c7">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E31">A6</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">H1</font> &amp; <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )
 ;<br/>

<a NAME="E2:31_1"/><b>then </b><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:31_1"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <b>by </b><i><a class="txt" href="msualg_3.html#E2:31_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E5:31_1"/>
<font color="Maroon" title="c8">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E3:31"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="msualg_3.html#E3:31_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="msualg_3.html#D2" target="_self" title="MSUALG_3:def.2">Def2</a></i>;<br/>
<a NAME="E6:31_1"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#T62" title="FUNCT_1:th.62">FUNCT_1:62</a></i>;<br/>
<b>hence </b><a NAME="E7:31_1"/>
<font color="Maroon" title="c7">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="msualg_3.html#E1:31"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="msualg_3.html#E3:31"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="msualg_3.html#E4:31"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="msualg_3.html#E1:31_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="msualg_3.html#E2:31_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="msualg_3.html#D5" target="_self" title="MSUALG_3:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:31"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:31_2"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ;<br/>

<a NAME="E2:31_2"/><b>then </b><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">H</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <b>by </b><i><a class="txt" href="msualg_3.html#E1:31_2"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E4:31_2"/>
<font color="Maroon" title="c7">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E3:31"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="msualg_3.html#E2:31_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="msualg_3.html#D2" target="_self" title="MSUALG_3:def.2">Def2</a></i>;<br/>
<a NAME="E5:31_2"/><b>then </b><i><font color="Green" title="E33">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
<a NAME="E6:31_2"/>
( the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="msualg_3.html#E1:31_2"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>
<a NAME="E7:31_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E5:31_2"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:31_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="msualg_3.html#E1:31"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="msualg_3.html#E3:31"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="msualg_3.html#E4:31"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="msualg_3.html#E1:31_2"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="msualg_3.html#D5" target="_self" title="MSUALG_3:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:31"/><b>then </b><i><font color="Green" title="E31">A12</font></i>: 
<font color="Maroon" title="c5">H1</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D3" target="_self" title="MSUALG_3:def.3">Def3</a></i>;<br/>
<a NAME="E8:31"/><i><font color="Green" title="E32">A13</font></i>: 
<font color="Maroon" title="c5">H1</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E5:31"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="msualg_3.html#D2" target="_self" title="MSUALG_3:def.2">Def2</a></i>;<br/>
<a NAME="E9:31"/><i><font color="Green" title="E33">A14</font></i>: 
<font color="Maroon" title="c5">H1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E1:31"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="msualg_3.html#E27"><i><font color="Green" title="E24">Lm2</font></i></a></i>;<br/>
<a NAME="E10:31"/><b>then </b><i><font color="Green" title="E34">A15</font></i>: 
<font color="Maroon" title="c5">H1</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E7:31"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="msualg_3.html#D10" target="_self" title="MSUALG_3:def.10">Def10</a></i>;<br/>
<a NAME="E11:31"/>
<font color="Maroon" title="c5">H1</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E8:31"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="msualg_3.html#E9:31"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="msualg_3.html#D11" target="_self" title="MSUALG_3:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E12:31"/>
<font color="Maroon" title="c5">H1</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c2">U1</font>
 <b>by </b><i><a class="txt" href="msualg_3.html#E10:31"><i><font color="Green" title="E34">A15</font></i></a>, <a class="ref" href="msualg_3.html#D12" target="_self" title="MSUALG_3:def.12">Def12</a></i>;<br/>


</div>
