
theorem Th6:
  for X be non empty set, A be set, F,G be FinSequence of X st dom
G = dom F & (for i be Nat st i in dom G holds G.i = A /\ F.i) holds union rng G
  = A /\ union rng F
proof
  let X be non empty set;
  let A be set;
  let F,G be FinSequence of X;
  assume that
A1: dom G = dom F and
A2: for i be Nat st i in dom G holds G.i = A /\ F.i;
  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
A3: r in E and
A4: E in rng G by TARSKI:def 4;
    consider s being object such that
A5: s in dom G and
A6: E = G.s by A4,FUNCT_1:def 3;
    reconsider s as Element of NAT by A5;
A7: r in A /\ F.s by A2,A3,A5,A6;
    then
A8: r in F.s by XBOOLE_0:def 4;
    F.s in rng F by A1,A5,FUNCT_1:3;
    then
A9: r in union rng F by A8,TARSKI:def 4;
    r in A by A7,XBOOLE_0:def 4;
    hence thesis by A9,XBOOLE_0:def 4;
  end;
  let r be object;
  assume
A10: r in A /\ union rng F;
  then
A11: r in A by XBOOLE_0:def 4;
  r in union rng F by A10,XBOOLE_0:def 4;
  then consider E being set such that
A12: r in E and
A13: E in rng F by TARSKI:def 4;
  consider s being object such that
A14: s in dom F and
A15: E = F.s by A13,FUNCT_1:def 3;
  reconsider s as Element of NAT by A14;
  A /\ E = G.s by A1,A2,A14,A15;
  then
A16: r in G.s by A11,A12,XBOOLE_0:def 4;
  G.s in rng G by A1,A14,FUNCT_1:3;
  hence thesis by A16,TARSKI:def 4;
end;
