
theorem Th7:
  for T being non empty TopSpace, A being Subset of T st A is open
  for C being Subset of T st C in Indiscernible(T) & C meets A holds C c= A
proof
  let T be non empty TopSpace;
  let A be Subset of T such that
A1: A is open;
  set R = Indiscernibility(T);
  let C be Subset of T;
  assume that
A2: C in Indiscernible(T) and
A3: C meets A;
  consider y being object such that
A4: y in C and
A5: y in A by A3,XBOOLE_0:3;
  consider x being object such that
  x in the carrier of T and
A6: C = Class(R,x) by A2,EQREL_1:def 3;
  for p being object st p in C holds p in A
  proof
    let p be object;
    [y,x] in R by A6,A4,EQREL_1:19;
    then
A7: [x,y] in R by EQREL_1:6;
    assume
A8: p in C;
    then [p,x] in R by A6,EQREL_1:19;
    then [p,y] in R by A7,EQREL_1:7;
    hence thesis by A1,A5,A8,Def3;
  end;
  hence thesis by TARSKI:def 3;
end;
