reserve i,j,k,n,m for Nat,
  x,y,z,y1,y2 for object, X,Y,D for set,
  p,q for XFinSequence;
reserve k1,k2 for Nat;

theorem Th25:
  for A,B1,B2 being set st B1 <N< B2 holds A/\ B1 <N< A/\B2
proof
  let A,B1,B2 be set;
  assume
A1: B1 <N< B2;
  for n,m st n in A/\B1 & m in A/\B2 holds n<m
  proof
    let n,m;
    assume that
A2: n in A/\B1 and
A3: m in A/\B2;
A4: m in B2 by A3,XBOOLE_0:def 4;
    n in B1 by A2,XBOOLE_0:def 4;
    hence thesis by A1,A4;
  end;
  hence thesis;
end;
