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 Th32:
  for X1, X2, Y1, Y2 being non empty SubSpace of X st X1,Y1
constitute_a_decomposition & X2,Y2 constitute_a_decomposition holds Y1 union Y2
  = the TopStruct of X iff X1 misses X2
proof
  let X1, X2, Y1, Y2 be non empty SubSpace of X;
  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 that
A1: X1,Y1 constitute_a_decomposition and
A2: X2,Y2 constitute_a_decomposition;
A3: A2,B2 constitute_a_decomposition by A2;
  then
A4: A2 = B2` by Th3;
A5: A2 = B2` by A3,Th3;
A6: A1,B1 constitute_a_decomposition by A1;
  then
A7: A1 = B1` by Th3;
  thus Y1 union Y2 = the TopStruct of X implies X1 misses X2
  proof
    assume Y1 union Y2 = the TopStruct of X;
    then B1 \/ B2 = the carrier of X by TSEP_1:def 2;
    then (B1 \/ B2)` = {}X by XBOOLE_1:37;
    then A1 /\ A2 = {} by A7,A5,XBOOLE_1:53;
    then A1 misses A2;
    hence thesis;
  end;
  assume X1 misses X2;
  then A1 misses A2;
  then
A8: A1 /\ A2 = {}X;
  A1 = B1` by A6,Th3;
  then (B1 \/ B2)` = {}X by A4,A8,XBOOLE_1:53;
  then B1 \/ B2 = ({}X)`;
  then
A9: the carrier of Y1 union Y2 = the carrier of X by TSEP_1:def 2;
  X is SubSpace of X by TSEP_1:2;
  hence thesis by A9,TSEP_1:5;
end;
