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 Th10:
  (for n holds B.n = A) implies Union B = A
proof
  assume
A1: for n holds B.n = A;
  now
    let x be object;
    assume x in rng B;
    then ex n st x = B.n by Th4;
    hence x = A by A1;
  end;
  then rng B = {A} by ZFMISC_1:35;
  then union rng B = A by ZFMISC_1:25;
  hence thesis by CARD_3:def 4;
end;
