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 holds A3.n = A1.n /\ A2.n) implies for n holds (
  superior_setsequence A3).n c= (superior_setsequence A1).n /\ (
  superior_setsequence A2).n
proof
  assume
A1: for n holds A3.n = A1.n /\ A2.n;
  let n;
  reconsider X3 = superior_setsequence A3 as SetSequence of X;
  reconsider X2 = superior_setsequence A2 as SetSequence of X;
  set B = A1;
  reconsider X1 = superior_setsequence B as SetSequence of X;
  X3.n c= X1.n /\ X2.n
  proof
    let x be object;
    assume x in X3.n;
    then consider k being Nat such that
A2: x in A3.(n+k) by Th20;
A3: A3.(n+k) = B.(n+k) /\ A2.(n+k) by A1;
    then x in A2.(n+k) by A2,XBOOLE_0:def 4;
    then
A4: x in X2.n by Th20;
    x in B.(n+k) by A2,A3,XBOOLE_0:def 4;
    then x in X1.n by Th20;
    hence thesis by A4,XBOOLE_0:def 4;
  end;
  hence thesis;
end;
