reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X0 for non empty SubSpace of X,
  B1, B2 for Subset of X0;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;

theorem Th35:
  X1 meets Y1 & X1,X2 constitute_a_decomposition & Y1,Y2
  constitute_a_decomposition implies (X1 meet Y1),(X2 union Y2)
  constitute_a_decomposition
proof
  reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by
TSEP_1:1;
  reconsider B1 = the carrier of Y1, B2 = the carrier of Y2 as Subset of X by
TSEP_1:1;
  assume
A1: X1 meets Y1;
  assume for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the
  carrier of X2 holds A1,A2 constitute_a_decomposition;
  then
A2: A1,A2 constitute_a_decomposition;
  assume for B1, B2 being Subset of X st B1 = the carrier of Y1 & B2 = the
  carrier of Y2 holds B1,B2 constitute_a_decomposition;
  then
A3: B1,B2 constitute_a_decomposition;
  now
    let C, D be Subset of X;
    assume C = the carrier of X1 meet Y1 & D = the carrier of X2 union Y2;
    then C = A1 /\ B1 & D = A2 \/ B2 by A1,TSEP_1:def 2,def 4;
    hence C,D constitute_a_decomposition by A2,A3,Th13;
  end;
  hence thesis;
end;
