let G1 be _Graph; :: thesis: for G2 being Subgraph of G1
for x, y, e being set holds
( ( e Joins x,y,G2 implies e Joins x,y,G1 ) & ( e DJoins x,y,G2 implies e DJoins x,y,G1 ) & ( e SJoins x,y,G2 implies e SJoins x,y,G1 ) & ( e DSJoins x,y,G2 implies e DSJoins x,y,G1 ) )
let G2 be Subgraph of G1; :: thesis: for x, y, e being set holds
( ( e Joins x,y,G2 implies e Joins x,y,G1 ) & ( e DJoins x,y,G2 implies e DJoins x,y,G1 ) & ( e SJoins x,y,G2 implies e SJoins x,y,G1 ) & ( e DSJoins x,y,G2 implies e DSJoins x,y,G1 ) )
let x, y, e be set ; :: thesis: ( ( e Joins x,y,G2 implies e Joins x,y,G1 ) & ( e DJoins x,y,G2 implies e DJoins x,y,G1 ) & ( e SJoins x,y,G2 implies e SJoins x,y,G1 ) & ( e DSJoins x,y,G2 implies e DSJoins x,y,G1 ) )
thus ( e Joins x,y,G2 implies e Joins x,y,G1 ) by Lm5; :: thesis: ( ( e DJoins x,y,G2 implies e DJoins x,y,G1 ) & ( e SJoins x,y,G2 implies e SJoins x,y,G1 ) & ( e DSJoins x,y,G2 implies e DSJoins x,y,G1 ) )

hereby :: thesis: ( ( e SJoins x,y,G2 implies e SJoins x,y,G1 ) & ( e DSJoins x,y,G2 implies e DSJoins x,y,G1 ) )

assume e DJoins x,y,G2 ; :: thesis: e DJoins x,y,G1
then A1: ( e in the_Edges_of G2 & (the_Source_of G2) . e = x & (the_Target_of G2) . e = y ) by Def16;
then ( (the_Source_of G1) . e = x & (the_Target_of G1) . e = y ) by Def34;
hence e DJoins x,y,G1 by A1, Def16; :: thesis: verum

end;

hereby :: thesis: ( e DSJoins x,y,G2 implies e DSJoins x,y,G1 )

assume e SJoins x,y,G2 ; :: thesis: e SJoins x,y,G1
then A2: ( e in the_Edges_of G2 & ( ( (the_Source_of G2) . e in x & (the_Target_of G2) . e in y ) or ( (the_Source_of G2) . e in y & (the_Target_of G2) . e in x ) ) ) by Def17;
then ( (the_Source_of G2) . e = (the_Source_of G1) . e & (the_Target_of G2) . e = (the_Target_of G1) . e ) by Def34;
hence e SJoins x,y,G1 by A2, Def17; :: thesis: verum

end;

assume e DSJoins x,y,G2 ; :: thesis: e DSJoins x,y,G1
then A3: ( e in the_Edges_of G2 & (the_Source_of G2) . e in x & (the_Target_of G2) . e in y ) by Def18;
then ( (the_Source_of G2) . e = (the_Source_of G1) . e & (the_Target_of G2) . e = (the_Target_of G1) . e ) by Def34;
hence e DSJoins x,y,G1 by A3, Def18; :: thesis: verum