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
  (for n being Nat holds A3.n = A1.n /\ A2.n) implies
 for n being Nat holds (
  inferior_setsequence A3).n = (inferior_setsequence A1).n /\ (
  inferior_setsequence A2).n
proof
  assume
A1: for n being Nat holds A3.n = A1.n /\ A2.n;
  let n be Nat;
  reconsider X3 = inferior_setsequence A3 as SetSequence of X;
  reconsider X2 = inferior_setsequence A2 as SetSequence of X;
  set B = A1;
  reconsider X1 = inferior_setsequence B as SetSequence of X;
A2: X1.n /\ X2.n c= X3.n
  proof
    let x be object;
    assume x in (X1.n /\ X2.n);
    then
A3: x in X1.n & x in X2.n by XBOOLE_0:def 4;
    now
      let k be Nat;
      x in B.(n+k) & x in A2.(n+k) by A3,Th19;
      then x in B.(n+k) /\ A2.(n+k) by XBOOLE_0:def 4;
      hence x in A3.(n+k) by A1;
    end;
    hence thesis by Th19;
  end;
  X3.n c= X1.n /\ X2.n
  proof
    let x be object;
    assume
A4: x in X3.n;
    now
      let k be Nat;
      x in A3.(n+k) by A4,Th19;
      then x in (B.(n+k) /\ A2.(n+k)) by A1;
      hence x in B.(n+k) & x in A2.(n+k) by XBOOLE_0:def 4;
    end;
    then x in X1.n & x in X2.n by Th19;
    hence thesis by XBOOLE_0:def 4;
  end;
  hence thesis by A2,XBOOLE_0:def 10;
end;
