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<b>let </b><font color="Maroon" title="c2">F1</font>, <font color="Maroon" title="c3">F2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</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">u</font> being   <a href="unialg_3.html#M2" title="UNIALG_3:mode.2">Element</a> of  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font> holds  <font color="Maroon" title="c2">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_3.html#K4" title="UNIALG_3:func.4">carr</a> <font color="Olive" title="b1">u</font> ) &amp; ( for <font color="Olive" title="b1">u</font> being   <a href="unialg_3.html#M2" title="UNIALG_3:mode.2">Element</a> of  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font> holds  <font color="Maroon" title="c3">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_3.html#K4" title="UNIALG_3:func.4">carr</a> <font color="Olive" title="b1">u</font> ) implies <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6_1_2"/><i><font color="Green" title="E3">A1</font></i>: 
for <font color="Olive" title="b1">u1</font> being   <a href="unialg_3.html#M2" title="UNIALG_3:mode.2">Element</a> of  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font> holds  <font color="Maroon" title="c2">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_3.html#K4" title="UNIALG_3:func.4">carr</a> <font color="Olive" title="b1">u1</font>
 <b>and </b><br/><a NAME="E2:6_1_2"/><i><font color="Green" title="E4">A2</font></i>: 
for <font color="Olive" title="b1">u2</font> being   <a href="unialg_3.html#M2" title="UNIALG_3:mode.2">Element</a> of  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font> holds  <font color="Maroon" title="c3">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_3.html#K4" title="UNIALG_3:func.4">carr</a> <font color="Olive" title="b1">u2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font></span><br/>

<a NAME="E3:6_1_2"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font> holds <br/><font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font> implies <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font> )</span><br/>

<b>assume </b><a NAME="E1:6_1_2_1"/>
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">u1</font> = <font color="Maroon" title="c4">u</font> as    <a href="unialg_3.html#M2" title="UNIALG_3:mode.2">Element</a> of  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">U1</font> being    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font><b> such that </b><br/><a NAME="E4:6_1_2_1"/><i><font color="Green" title="E5">A3</font></i>: 
( <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">U1</font> &amp;  <a href="unialg_3.html#K4" title="UNIALG_3:func.4">carr</a> <font color="Maroon" title="c5">u1</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="c6">U1</font> )
 <b>by </b><i><a class="ref" href="unialg_3.html#D2" target="_self" title="UNIALG_3:def.2">Def2</a></i>;<br/>
<a NAME="E5:6_1_2_1"/>
<font color="Maroon" title="c2">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">u1</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="c6">U1</font>
 
<b>by </b><i><a class="txt" href="#E1:6_1_2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="#E4:6_1_2_1"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:6_1_2_1"/>
<font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font>
 <b>by </b><i><a class="txt" href="#E2:6_1_2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="#E4:6_1_2_1"><i><font color="Green" title="E5">A3</font></i></a></i>; <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>
<b>hence </b><a NAME="E4:6_1_2"/>
<font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</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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