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<b>let </b><font color="Maroon" title="c1">U1</font>, <font color="Maroon" title="c2">U2</font>, <font color="Maroon" title="c3">U3</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font><br/> for <font color="Olive" title="b2">h1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font>  st <font color="Olive" title="b1">h</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Olive" title="b2">h1</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font> holds <br/><font color="Olive" title="b2">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c3">U3</font></span><br/><b>let </b><font color="Maroon" title="c4">h</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">h1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font>  st <font color="Maroon" title="c4">h</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Olive" title="b1">h1</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font> holds <br/><font color="Olive" title="b1">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c3">U3</font></span><br/><b>let </b><font color="Maroon" title="c5">h1</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">h</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c5">h1</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font> implies <font color="Maroon" title="c5">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c3">U3</font> )</span><br/>









<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c4">h</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c5">h1</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c3">U3</font></span><br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c4">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c5">h1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="alg_1.html#T8" target="_self" title="ALG_1:th.8">Th8</a></i>;<br/>
<a NAME="E3:15"/><i><font color="Green" title="E19">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font></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="c3">U3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1"/><i><font color="Green" title="E20">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h1</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">U2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:15_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h</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">U2</font>
 
<b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="alg_1.html#T9" target="_self" title="ALG_1:th.9">Th9</a></i>;<br/>
<b>hence </b> <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font></span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">h1</font>
<b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U3</font>
<b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="alg_1.html#T9" target="_self" title="ALG_1:th.9">Th9</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:15"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c5">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</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="#E2:15"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>
<a NAME="E5:15"/>
( <font color="Maroon" title="c4">h</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c5">h1</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font> )
 
<b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="alg_1.html#T8" target="_self" title="ALG_1:th.8">Th8</a></i>;<br/>
<a NAME="E6:15"/><b>then </b>
<font color="Maroon" title="c5">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c3">U3</font>
 
<b>by </b><i><a class="ref" href="alg_1.html#T7" target="_self" title="ALG_1:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E7:15"/>
<font color="Maroon" title="c5">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c3">U3</font>
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="alg_1.html#T8" target="_self" title="ALG_1:th.8">Th8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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