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
  n1 <> n2 implies n1 in dom GS.set(n1,x).set(n2,y) & n2 in dom GS.set(
n1,x).set(n2,y) & GS.set(n1,x).set(n2,y).n1 = x & GS.set(n1,x).set(n2,y).n2 = y
proof
  assume
A1: n1 <> n2;
  set G2 = GS.set(n1,x), G3 = G2.set(n2,y);
A2: dom G3 = dom G2 \/ {n2} by Th7;
  dom G2 = dom GS \/ {n1} & n1 in {n1} by Th7,TARSKI:def 1;
  then n1 in dom G2 by XBOOLE_0:def 3;
  hence n1 in dom GS.set(n1,x).set(n2,y) by A2,XBOOLE_0:def 3;
  n2 in {n2} by TARSKI:def 1;
  hence n2 in dom GS.set(n1,x).set(n2,y) by A2,XBOOLE_0:def 3;
  thus GS.set(n1,x).set(n2,y).n1 = G2.n1 by A1,Th9
    .= x by Th8;
  thus thesis by Th8;
end;
