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 Th95:
  for G1,G2 being _Graph, V,E being set, G3 being inducedSubgraph of G1,
  V,E st G1 == G2 holds G3 is inducedSubgraph of G2,V,E
proof
  let G1,G2 be _Graph, V,E be set, G3 be inducedSubgraph of G1,V,E;
  assume
A1: G1 == G2;
  now
    per cases;
    suppose
A2:   V is non empty Subset of the_Vertices_of G1 & E c= G1 .edgesBetween(V);
      then
A3:   the_Vertices_of G3 = V & the_Edges_of G3 = E by Def37;
A4:   G3 is Subgraph of G2 by A1,Th91;
      V is non empty Subset of the_Vertices_of G2 & E c= G2.edgesBetween(V
      ) by A1,A2,Th90;
      hence thesis by A3,A4,Def37;
    end;
    suppose
A5:   not (V is non empty Subset of the_Vertices_of G1 & E c= G1
      .edgesBetween(V));
      then
A6:   not (V is non empty Subset of the_Vertices_of G2 & E c= G2
      .edgesBetween(V)) by A1,Th90;
      G3 == G1 by A5,Def37;
      then
A7:   G3 == G2 by A1;
      then G3 is Subgraph of G2 by Th87;
      hence thesis by A6,A7,Def37;
    end;
  end;
  hence thesis;
end;
