
theorem Th10:
  for F being finite set, A being FinSequence of bool F
  for i being Element of NAT
  for x,y being set st x <> y & x in A.i & y in A.i holds
    (A.i \ {x}) \/ (A.i \ {y}) = A.i
proof
  let F be finite set;
  let A be FinSequence of bool F;
  let i be Element of NAT;
  let x,y be set such that
A1: x <> y and
A2: x in A.i and
A3: y in A.i;
  A.i c= (A.i\{x}) \/ (A.i\{y})
  proof
    {} = {y} \ ({y} \/ {}) by XBOOLE_1:46;
    then A.i = A.i \ ({y} \ {y});
    then A.i = (A.i \ {y}) \/ A.i /\ {y} by XBOOLE_1:52;
    then
A4: A.i = A.i \ {y} \/ {y} by A3,ZFMISC_1:46;
    let z be object;
    not x in {y} by A1,TARSKI:def 1;
    then
A5: x in A.i\{y} by A2,XBOOLE_0:def 5;
    assume z in A.i;
    then z in (A.i \ {x} \/ {x}) \/ (A.i \ {y} \/ {y}) by A4,XBOOLE_0:def 3;
    then z in (A.i \ {x}) \/ ({x} \/ ({y} \/ (A.i \{y}))) by XBOOLE_1:4;
    then z in (A.i \ {x}) \/ (({x} \/ {y}) \/ (A.i \{y})) by XBOOLE_1:4;
    then z in ((A.i \ {x}) \/ ({y} \/ {x})) \/ (A.i \{y}) by XBOOLE_1:4;
    then z in (((A.i \ {x}) \/ {y}) \/ {x}) \/ (A.i \{y}) by XBOOLE_1:4;
    then
A6: z in ((A.i \ {x}) \/ {y}) \/ ({x} \/ (A.i \{y})) by XBOOLE_1:4;
    not y in {x} by A1,TARSKI:def 1;
    then y in A.i\{x} by A3,XBOOLE_0:def 5;
    then z in (A.i \ {x}) \/ ({x} \/ (A.i \{y})) by A6,ZFMISC_1:40;
    hence thesis by A5,ZFMISC_1:40;
  end;
  hence thesis;
end;
