reserve GS for GraphStruct;
reserve G,G1,G2,G3 for _Graph;
reserve e,x,x1,x2,y,y1,y2,E,V,X,Y for set;
reserve n,n1,n2 for Nat;
reserve v,v1,v2 for Vertex of G;

theorem Th72:
  for G1 being _Graph, G2 being Subgraph of G1,
      x,y being set, e being object
  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)
proof
  let G1 be _Graph, G2 be Subgraph of G1, x,y be set, e be object;
  thus e Joins x,y,G2 implies e Joins x,y,G1 by Lm4;
  hereby
    assume
A1: e DJoins x,y,G2;
    then
A2: e in the_Edges_of G2;
    (the_Target_of G2) .e=y by A1;
    then
A3: (the_Target_of G1).e = y by A2,Def32;
    (the_Source_of G2).e=x by A1;
    then (the_Source_of G1).e = x by A2,Def32;
    hence e DJoins x,y,G1 by A2,A3;
  end;
  hereby
    assume
A4: e SJoins x,y,G2;
    then
A5: (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;
A6: e in the_Edges_of G2 by A4;
    then (the_Source_of G2).e = (the_Source_of G1).e & (the_Target_of G2).e =
    ( the_Target_of G1).e by Def32;
    hence e SJoins x,y,G1 by A6,A5;
  end;
  assume
A7: e DSJoins x,y,G2;
  then
A8: (the_Source_of G2).e in x & (the_Target_of G2).e in y;
A9: e in the_Edges_of G2 by A7;
  then (the_Source_of G2).e = (the_Source_of G1).e & (the_Target_of G2).e = (
  the_Target_of G1).e by Def32;
  hence thesis by A9,A8;
end;
