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 Th10:
  for X1,X2 being non empty Subset of S-Terms V holds
  (X1 \/ X2)-CircuitStr = (X1-CircuitStr)+*(X2-CircuitStr)
proof
  let X1,X2 be non empty Subset of S-Terms V;
  set X = X1 \/ X2;
  set C = [:the carrier' of S,{the carrier of S}:];
A1: Subtrees X = (Subtrees X1) \/ Subtrees X2 by Th7;
A2: C-Subtrees X = (C-Subtrees X1) \/ (C-Subtrees X2) by Th8;
  (for t being Element of X1 holds t is finite)
  & for t being Element of X2 holds t is finite;
  then
A3: C-ImmediateSubtrees X = (C-ImmediateSubtrees X1)+*(C
  -ImmediateSubtrees X2) by Th9;
  id (C-Subtrees X) = (id (C-Subtrees X1))+*id (C-Subtrees X2)
  by A2,FUNCT_4:22;
  hence thesis by A1,A2,A3,CIRCCOMB:def 2;
end;
