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 Th7:
  for X1,X2 being constituted-DTrees non empty set holds
  Subtrees (X1 \/ X2) = (Subtrees X1) \/ (Subtrees X2)
proof
  let X1,X2 be constituted-DTrees non empty set;
  hereby
    let x be object;
    assume x in Subtrees (X1 \/ X2);
    then consider t being Element of X1 \/ X2, n being Node of t such that
A1: x = t|n by TREES_9:19;
    t in X1 or t in X2 by XBOOLE_0:def 3;
    then x in Subtrees X1 or x in Subtrees X2 by A1,TREES_9:19;
    hence x in (Subtrees X1) \/ (Subtrees X2) by XBOOLE_0:def 3;
  end;
  let x be object;
  assume
A2: x in (Subtrees X1) \/ (Subtrees X2);
  per cases by A2,XBOOLE_0:def 3;
  suppose x in Subtrees X1;
    then consider t being Element of X1, n being Node of t such that
A3: x = t|n by TREES_9:19;
    t is Element of X1 \/ X2 by XBOOLE_0:def 3;
    hence thesis by A3,TREES_9:19;
  end;
  suppose x in Subtrees X2;
    then consider t being Element of X2, n being Node of t such that
A4: x = t|n by TREES_9:19;
    t is Element of X1 \/ X2 by XBOOLE_0:def 3;
    hence thesis by A4,TREES_9:19;
  end;
end;
