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
  for G being non-multi Graph st G1 c= G holds G1 is non-multi
proof
  let G be non-multi Graph;
  assume G1 c= G;
then A1: G1 is Subgraph of G;
   for x,y being set st x in the carrier' of G1 & y in the carrier' of G1 &
  ( (the Source of G1).x = (the Source of G1).y &
  (the Target of G1).x = (the Target of G1).y or
  (the Source of G1).x = (the Target of G1).y &
  (the Source of G1).y = (the Target of G1).x ) holds x = y
  proof
    let x,y be set such that
A2: x in the carrier' of G1 & y in the carrier' of G1;
    assume
A3: (the Source of G1).x = (the Source of G1).y &
    (the Target of G1).x = (the Target of G1).y or
    (the Source of G1).x = (the Target of G1).y &
    (the Source of G1).y = (the Target of G1).x;
A4: (the carrier' of G1) c= (the carrier' of G) by A1,Def18;
    A5: (
the Source of G1).x = (the Source of G).x & (the Source of G1).y = (the
    Source of G).y by A1,A2,Def18;
     (
the Target of G1).x = (the Target of G).x & (the Target of G1).y = (the
    Target of G).y by A1,A2,Def18;
    hence thesis by A2,A3,A4,A5,Def8;
  end;
  hence thesis;
end;
