let G1 be _Graph; :: thesis:
let G2 be Subgraph of G1; :: thesis:
let v1 be Vertex of G1; :: thesis:
let v2 be Vertex of G2; :: thesis:
let e be set ; :: thesis:
assume that
A1: v1 = v2 and
A2: e in the_Edges_of G2 ; :: thesis:
A3: ( (the_Source_of G2) . e = (the_Source_of G1) . e & (the_Target_of G2) . e = (the_Target_of G1) . e ) by A2, Def32;

per cases ) . e = v1 or ( (the_Source_of G1) . e = v1 & (the_Target_of G1) . e <> v1 ) or ( (the_Source_of G1) . e <> v1 & (the_Target_of G1) . e <> v1 ) ) ;

suppose A4: (the_Target_of G1) . e = v1 ; :: thesis:

hence v1 .adj e = (the_Source_of G1) . e by A2, Def41
.= v2 .adj e by A1, A2, A3, A4, Def41 ;
:: thesis:

end;

suppose A5: ( (the_Source_of G1) . e = v1 & (the_Target_of G1) . e <> v1 ) ; :: thesis:

hence v1 .adj e = (the_Target_of G1) . e by A2, Def41
.= v2 .adj e by A1, A2, A3, A5, Def41 ;
:: thesis:

end;

suppose A6: ( (the_Source_of G1) . e <> v1 & (the_Target_of G1) . e <> v1 ) ; :: thesis:

hence v1 .adj e = v2 by A1, Def41
.= v2 .adj e by A1, A3, A6, Def41 ;
:: thesis:

end;

end;

end;

hence v1 .adj e = v2 .adj e ; :: thesis: