reserve n,m,k,k1,k2,i,j for Nat;
reserve x,y,z for object,X,Y,Z for set;
reserve A for Subset of X;
reserve B,A1,A2,A3 for SetSequence of X;
reserve Si for SigmaField of X;
reserve S,S1,S2,S3 for SetSequence of Si;

theorem Th32:
  Complement (inferior_setsequence B) = (superior_setsequence Complement B)
proof
  reconsider A2 = inferior_setsequence B as SetSequence of X;
  reconsider A3 = superior_setsequence Complement B as SetSequence of X;
  now
    let n be Element of NAT;
    (A2.n)` = ((A3.n)`)` by Th30;
    hence (Complement A2).n = A3.n by PROB_1:def 2;
  end;
  hence thesis by FUNCT_2:63;
end;
