reserve x, y, z, v for set,
  n, m, k for Nat;
reserve G, G1, G2, G3 for Graph;
reserve x, y for Element of (the carrier of G);

theorem Th24:
  (the Source of G1) tolerates (the Source of G2) &
  (the Target of G1) tolerates (the Target of G2) &
  (G1 \/ G2 = G2 or G2 \/ G1 = G2) implies G1 c= G2
proof
  assume
  A1: (the Source of G1) tolerates (the Source of G2) & (the Target of G1)
  tolerates (the Target of G2);
  assume
A2: G1 \/ G2 = G2 or G2 \/ G1 = G2;
then
 the carrier of G2 = (the carrier of G1) \/ (the carrier of G2) by A1,Def5;
then A3: (the carrier of G1) c= the carrier of G2 by XBOOLE_1:7;
 the carrier' of G2 = (the carrier' of G1) \/ (the carrier' of G2)
  by A1,A2,Def5;
then A4: (the carrier' of G1) c= (the carrier' of G2) by XBOOLE_1:7;
 for v st v in the carrier' of G1 holds
  (the Source of G1).v = (the Source of G2).v &
  (the Target of G1).v = (the Target of G2).v &
  (the Source of G2).v in the carrier of G1 &
  (the Target of G2).v in the carrier of G1
  proof
    let v;
    assume
A5: v in the carrier' of G1;
    thus
then A6: (the Source of G1).v = (the Source of G2).v by A1,A2,Def5;
    thus
A7: (the Target of G1).v = (the Target of G2).v by A1,A2,A5,Def5;
    thus (the Source of G2).v in the carrier of G1 by A5,A6,FUNCT_2:5;
    thus thesis by A5,A7,FUNCT_2:5;
  end;
then  G1 is Subgraph of G2 by A3,A4,Def18;
  hence thesis;
end;
