reserve
  A,B,X for set,
  S for SigmaField of X;

theorem Th1:
  for S being non empty Subset-Family of X, F, G being sequence of S,
  A being Element of S st for n being Element of NAT holds G.n = A /\ F.n
  holds union rng G = A /\ union rng F
proof
  let S be non empty Subset-Family of X;
  let F, G be sequence of S, A be Element of S;
  assume
A1: for n being Element of NAT holds G.n = A /\ F.n;
  thus union rng G c= A /\ union rng F
  proof
    let r be object;
    assume r in union rng G;
    then consider E being set such that
A2: r in E and
A3: E in rng G by TARSKI:def 4;
    consider s being object such that
A4: s in dom G and
A5: E = G.s by A3,FUNCT_1:def 3;
    reconsider s as Element of NAT by A4;
A6: r in A /\ F.s by A1,A2,A5;
    then
A7: r in A by XBOOLE_0:def 4;
A8: F.s in rng F by FUNCT_2:4;
    r in F.s by A6,XBOOLE_0:def 4;
    then r in union rng F by A8,TARSKI:def 4;
    hence thesis by A7,XBOOLE_0:def 4;
  end;
  let r be object;
  assume
A9: r in A /\ union rng F;
  then
A10: r in A by XBOOLE_0:def 4;
  r in union rng F by A9,XBOOLE_0:def 4;
  then consider E being set such that
A11: r in E and
A12: E in rng F by TARSKI:def 4;
  consider s being object such that
A13: s in dom F and
A14: E = F.s by A12,FUNCT_1:def 3;
  reconsider s as Element of NAT by A13;
A15: G.s in rng G by FUNCT_2:4;
  A /\ E = G.s by A1,A14;
  then r in G.s by A10,A11,XBOOLE_0:def 4;
  hence thesis by A15,TARSKI:def 4;
end;
