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 Th18:
  G is_sum_of G1, G2 implies G1 c= G & G2 c= G
proof
  assume
A1: G is_sum_of G1, G2;
A2: for G, G1, G2 being Graph st G is_sum_of G1, G2 holds G1 c= G
  proof
    let G, G1, G2 be Graph;
    assume
A3: G is_sum_of G1, G2;
    then
A4: (the Source of G1) tolerates (the Source of G2) & (the Target of G1)
    tolerates (the Target of G2);
A5: the MultiGraphStruct of G = G1 \/ G2 by A3;
then  the carrier of G = (the carrier of G1) \/ (the carrier of G2)
    by A4,Def5;
then A6: (the carrier of G1) c= (the carrier of G) by XBOOLE_1:7;
 the carrier' of G = (the carrier' of G1) \/ (the carrier' of G2)
    by A4,A5,Def5;
then A7: (the carrier' of G1) c= (the carrier' of G) by XBOOLE_1:7;
 for v st v in the carrier' of G1 holds
    (the Source of G1).v = (the Source of G).v &
    (the Target of G1).v = (the Target of G).v &
    (the Source of G).v in the carrier of G1 &
    (the Target of G).v in the carrier of G1
    proof
      let v;
      assume
A8:  v in the carrier' of G1;
      thus
then A9:  (the Source of G1).v = (the Source of G).v by A4,A5,Def5;
      thus
A10:  (the Target of G1).v = (the Target of G).v by A4,A5,A8,Def5;
      thus (the Source of G).v in the carrier of G1 by A8,A9,FUNCT_2:5;
      thus thesis by A8,A10,FUNCT_2:5;
    end;
then  G1 is Subgraph of G by A6,A7,Def18;
    hence thesis;
  end;
  hence G1 c= G by A1;
  thus thesis by A1,A2,Th10;
end;
