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 Th29:
  X1,X2 constitute_a_decomposition iff X1 misses X2 & the
  TopStruct of X = X1 union X2
proof
  reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by
TSEP_1:1;
  thus X1,X2 constitute_a_decomposition implies X1 misses X2 & the TopStruct
  of X = X1 union X2
  proof
    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
A1: A1,A2 constitute_a_decomposition;
    then A1 misses A2;
    hence X1 misses X2;
    A1 \/ A2 = the carrier of X by A1;
    then
A2: the carrier of X1 union X2 = the carrier of X by TSEP_1:def 2;
    X is SubSpace of X by TSEP_1:2;
    hence thesis by A2,TSEP_1:5;
  end;
  assume
A3: X1 misses X2;
  assume the TopStruct of X = X1 union X2;
  then for A1, A2 be Subset of X st
  A1 = the carrier of X1 & A2 = the carrier of X2
  holds A1,A2 constitute_a_decomposition by A3,TSEP_1:def 2;
  hence thesis;
end;
