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
  (ex G st G1 c= G & G2 c= G & G3 c= G) implies (G1 \/ G2) \/ G3 = G1 \/ ( G2
  \/ G3)
proof
  given G such that
A1: G1 c= G and
A2: G2 c= G and
A3: G3 c= G;
A4: (the Source of G1) c= (the Source of G) by A1,Th5;
A5: (the Source of G2) c= (the Source of G) by A2,Th5;
A6: (the Source of G3) c= (the Source of G) by A3,Th5;
A7: (the Target of G1) c= (the Target of G) by A1,Th5;
A8: (the Target of G2) c= (the Target of G) by A2,Th5;
A9: (the Target of G3) c= (the Target of G) by A3,Th5;
A10: (the Source of G1) tolerates (the Source of G2) by A4,A5,PARTFUN1:57;
A11: (the Source of G1) tolerates (the Source of G3) by A4,A6,PARTFUN1:57;
A12: (the Source of G2) tolerates (the Source of G3) by A5,A6,PARTFUN1:57;
A13: (the Target of G1) tolerates (the Target of G2) by A7,A8,PARTFUN1:57;
A14: (the Target of G1) tolerates (the Target of G3) by A7,A9,PARTFUN1:57;
 (the Target of G2) tolerates (the Target of G3) by A8,A9,PARTFUN1:57;
  hence thesis by A10,A11,A12,A13,A14,Th9;
end;
