reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem Th37:
  Intersection (A (\+\) A1) c= A \+\ Intersection A1
proof
  let x be object;
  assume
A1: x in Intersection (A (\+\) A1);
A2: now
    let n;
    x in (A (\+\) A1).n by A1,PROB_1:13;
    then x in A \+\ A1.n by Def9;
    hence x in A & not x in A1.n or not x in A & x in A1.n by XBOOLE_0:1;
  end;
  assume not x in A \+\ Intersection A1;
  then
A3: not x in A & not x in Intersection A1 or x in Intersection A1 & x in A
  by XBOOLE_0:1;
  per cases by A3,PROB_1:13;
  suppose
    not x in A & not for n holds x in A1.n;
    hence contradiction by A2;
  end;
  suppose
A4: x in A & for n holds x in A1.n;
    then x in A1.0;
    hence contradiction by A2,A4;
  end;
end;
