reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;

theorem Th29:
  X1 meets X2 implies (X1 is SubSpace of X0 implies (X0 meet X2)
meets X1 & (X2 meet X0) meets X1) & (X2 is SubSpace of X0 implies (X1 meet X0)
  meets X2 & (X0 meet X1) meets X2)
proof
  assume
A1: X1 meets X2;
A2: now
    X1 meet X2 is SubSpace of X2 by A1,TSEP_1:27;
    then
A3: (X1 meet X2) meets X2 by Th17;
    assume
A4: X2 is SubSpace of X0;
    then X1 meet X2 is SubSpace of X1 meet X0 by A1,Th27;
    hence
A5: (X1 meet X0) meets X2 by A3,Th18;
    X1 meets X0 by A1,A4,Th18;
    hence (X0 meet X1) meets X2 by A5,TSEP_1:26;
  end;
  now
    X1 meet X2 is SubSpace of X1 by A1,TSEP_1:27;
    then
A6: (X1 meet X2) meets X1 by Th17;
    assume
A7: X1 is SubSpace of X0;
    then X1 meet X2 is SubSpace of X0 meet X2 by A1,Th27;
    hence
A8: (X0 meet X2) meets X1 by A6,Th18;
    X0 meets X2 by A1,A7,Th18;
    hence (X2 meet X0) meets X1 by A8,TSEP_1:26;
  end;
  hence thesis by A2;
end;
