reserve S for non empty non void ManySortedSign,
  V for non-empty ManySortedSet of the carrier of S,
  A for non-empty MSAlgebra over S,
  X for non empty Subset of S-Terms V,
  t for Element of X;

theorem Th8:
  for X1,X2 being constituted-DTrees non empty set, C be set holds
  C-Subtrees (X1 \/ X2) = (C-Subtrees X1) \/ (C-Subtrees X2)
proof
  let X1,X2 be constituted-DTrees non empty set, C be set;
  hereby
    let x be object;
    assume x in C-Subtrees (X1 \/ X2);
    then consider t being Element of X1 \/ X2, n being Node of t such that
A1: x = t|n and
A2: not n in Leaves dom t or t.n in C by TREES_9:24;
    t in X1 or t in X2 by XBOOLE_0:def 3;
    then x in C-Subtrees X1 or x in C-Subtrees X2 by A1,A2,TREES_9:24;
    hence x in (C-Subtrees X1) \/ (C-Subtrees X2) by XBOOLE_0:def 3;
  end;
  let x be object;
  assume
A3: x in (C-Subtrees X1) \/ (C-Subtrees X2);
  per cases by A3,XBOOLE_0:def 3;
  suppose x in C-Subtrees X1;
    then consider t being Element of X1, n being Node of t such that
A4: x = t|n and
A5: not n in Leaves dom t or t.n in C by TREES_9:24;
    t is Element of X1 \/ X2 by XBOOLE_0:def 3;
    hence thesis by A4,A5,TREES_9:24;
  end;
  suppose x in C-Subtrees X2;
    then consider t being Element of X2, n being Node of t such that
A6: x = t|n and
A7: not n in Leaves dom t or t.n in C by TREES_9:24;
    t is Element of X1 \/ X2 by XBOOLE_0:def 3;
    hence thesis by A6,A7,TREES_9:24;
  end;
end;
