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
  for S being non empty non void ManySortedSign
  for V being non-empty ManySortedSet of the carrier of S
  for X1,X2 being SetWithCompoundTerm of S,V
  for A being non-empty MSAlgebra over S holds
  (X1 \/ X2)-Circuit A = (X1-Circuit A)+*(X2-Circuit A)
proof
  let S be non empty non void ManySortedSign;
  let V be non-empty ManySortedSet of the carrier of S;
  let X1,X2 be SetWithCompoundTerm of S,V;
  consider t1 being CompoundTerm of S,V such that
A1: t1 in X1 by MSATERM:def 7;
  t1 in X1 \/ X2 by A1,XBOOLE_0:def 3;
  then reconsider X = X1 \/ X2 as SetWithCompoundTerm of S,V by MSATERM:def 7;
  let A be non-empty MSAlgebra over S;
  set C1 = X1-Circuit A, C2 = X2-Circuit A, C = X-Circuit A;
A2: C1 tolerates C2 by Th18;
  then
A3: the Sorts of C1 tolerates the Sorts of C2;
A4: the Charact of C1 tolerates the Charact of C2 by A2;
A5: the Sorts of C1+*C2 = (the Sorts of C1) +* (the Sorts of C2) by A3,
CIRCCOMB:def 4;
A6: the Charact of C1+*C2 = (the Charact of C1) +* (the Charact of C2) by A3,
CIRCCOMB:def 4;
A7: X-CircuitStr = X1-CircuitStr+*(X2-CircuitStr) by Th10;
A8: C1 tolerates C by Th18;
A9: C2 tolerates C by Th18;
A10: the Sorts of C1 tolerates the Sorts of C by A8;
A11: the Sorts of C2 tolerates the Sorts of C by A9;
A12: the Charact of C1 tolerates the Charact of C by A8;
A13: the Charact of C2 tolerates the Charact of C by A9;
A14: dom the Sorts of C1+*C2 = the carrier of X-CircuitStr by A7,PARTFUN1:def 2
;
  dom the Charact of C1+*C2 = the carrier' of X-CircuitStr by A7,PARTFUN1:def 2
;
  then
A15: dom the Charact of C1+*C2 = dom the Charact of C by PARTFUN1:def 2;
A16: dom the Sorts of C1+*C2 = dom the Sorts of C by A14,PARTFUN1:def 2;
A17: the Charact of C1+*C2 = the Charact of C by A4,A6,A12,A13,A15,FUNCT_4:125
,PARTFUN1:55;
  the Sorts of C1+*C2 = the Sorts of C by A3,A5,A10,A11,A16,FUNCT_4:125
,PARTFUN1:55;
  hence thesis by A17;
end;
