reserve n,m,k for Element of NAT,
  x,X for set,
  A1 for SetSequence of X,
  Si for SigmaField of X,
  XSeq for SetSequence of Si;
reserve Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem Th3:
  for A, B being Subset of X st A misses B holds (A,B) followed_by
  {} is disjoint_valued
proof
  let A,B be Subset of X;
  reconsider A1 = (A,B) followed_by {}X as SetSequence of X;
  assume
A1: A misses B;
  A1 is disjoint_valued
  proof
A2: A1.1 = B by FUNCT_7:123;
    let m,n be Nat such that
A3: m <> n;
A4: A1.0 = A by FUNCT_7:122;
    per cases;
    suppose
A5:   m = 0;
      then n >= 1 by A3,NAT_1:14;
      then
A6:   n + 1 > 1 by NAT_1:13;
      per cases by A6,NAT_1:22;
      suppose
        n > 1;
        then A1.n = {} by FUNCT_7:124;
        hence thesis;
      end;
      suppose
        n = 1;
        hence thesis by A1,A4,A5,FUNCT_7:123;
      end;
    end;
    suppose
A7:   m <> 0 & m = 1;
      n >= 1 or n <= 1;
      then
A8:   n + 1 > 1 or n < 1 + 1 by NAT_1:13;
      per cases by A3,A7,A8,NAT_1:14,22;
      suppose
        n > 1;
        then A1.n = {} by FUNCT_7:124;
        hence thesis;
      end;
      suppose
        n = 0;
        hence thesis by A1,A4,A2,A7;
      end;
    end;
    suppose
A9:   m <> 0 & m <> 1;
      then m >= 1 by NAT_1:14;
      then m +1 > 1 by NAT_1:13;
      then A1.m = {} by A9,FUNCT_7:124,NAT_1:22;
      hence thesis;
    end;
  end;
  hence thesis;
end;
