reserve i,j,k,n for Nat;
reserve x,x1,x2,x3,y1,y2,y3 for set;

theorem
  for n be Nat holds n-SuccRelStr is asymmetric
proof
  let n be Nat;
  set S = n-SuccRelStr;
  the InternalRel of S is_asymmetric_in field the InternalRel of S
  proof
    let x,y be object;
    assume that
    x in field the InternalRel of S and
    y in field the InternalRel of S and
A1: [x,y] in the InternalRel of S;
A2: [x,y] in {[i,i+1] where i is Element of NAT:i+1 < n} by A1,Def6;
    assume [y,x] in the InternalRel of S;
    then [y,x] in {[i9,i9+1] where i9 is Element of NAT:i9+1 < n} by Def6;
    then consider j be Element of NAT such that
A3: [y,x] = [j,j+1] and
    j+1<n;
A4: y = j & x = j+1 by A3,XTUPLE_0:1;
    consider i be Element of NAT such that
A5: [x,y] = [i,i+1] and
    i+1<n by A2;
    x = i & y = i+1 by A5,XTUPLE_0:1;
    hence contradiction by A4;
  end;
  hence the InternalRel of S is asymmetric;
end;
