let G1, G2, G3 be _Graph; :: thesis: ( G1 == G2 & G3 is Subgraph of G1 implies G3 is Subgraph of G2 )
assume that
A1: G1 == G2 and
A2: G3 is Subgraph of G1 ; :: thesis: G3 is Subgraph of G2
the_Vertices_of G3 c= the_Vertices_of G1 by A2, Def32;
hence the_Vertices_of G3 c= the_Vertices_of G2 by A1; :: according to GLIB_000:def 32 :: thesis: ( the_Edges_of G3 c= the_Edges_of G2 & ( for e being set st e in the_Edges_of G3 holds
( (the_Source_of G3) . e = (the_Source_of G2) . e & (the_Target_of G3) . e = (the_Target_of G2) . e ) ) )
the_Edges_of G3 c= the_Edges_of G1 by A2, Def32;
hence the_Edges_of G3 c= the_Edges_of G2 by A1; :: thesis: for e being set st e in the_Edges_of G3 holds
( (the_Source_of G3) . e = (the_Source_of G2) . e & (the_Target_of G3) . e = (the_Target_of G2) . e )
let e be set ; :: thesis: ( e in the_Edges_of G3 implies ( (the_Source_of G3) . e = (the_Source_of G2) . e & (the_Target_of G3) . e = (the_Target_of G2) . e ) )
assume A3: e in the_Edges_of G3 ; :: thesis: ( (the_Source_of G3) . e = (the_Source_of G2) . e & (the_Target_of G3) . e = (the_Target_of G2) . e )
hence (the_Source_of G3) . e = (the_Source_of G1) . e by A2, Def32
.= (the_Source_of G2) . e by A1 ;
:: thesis: (the_Target_of G3) . e = (the_Target_of G2) . e
thus (the_Target_of G3) . e = (the_Target_of G1) . e by A2, A3, Def32
.= (the_Target_of G2) . e by A1 ; :: thesis: verum