reserve i,j,k,k1,k2,n,m,i1,i2,j1,j2 for Element of NAT,
  x for set;

theorem Th2:
  for i1,i2 being Nat st i1,i2 are_adjacent & 1<=i1 & 1<=i2
   holds i1-'1,i2-'1 are_adjacent
proof
  let i1,i2 be Nat;
  assume that
A1: i1,i2 are_adjacent and
A2: 1<=i1 and
A3: 1<=i2;
  0<=i1-1 by A2,XREAL_1:48;
  then
A4: i1-'1=i1-1 by XREAL_0:def 2;
  0<=i2-1 by A3,XREAL_1:48;
  then
A5: i2-'1=i2-1 by XREAL_0:def 2;
  i2=i1+1 or i1=i2+1 by A1;
  then i2-1=i1-1+1 or i1-1=i2-1+1;
  hence thesis by A4,A5;
end;
