reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th41:
  for P1,S1,S2 be FinSequence-membered set holds
     P1^(S1\/S2) = P1^S1 \/ P1^S2
proof
  let P1,S1,S2 be FinSequence-membered set;
  thus P1^(S1\/S2) c= P1^S1 \/ P1^S2
  proof
    let x be object such that
A1:   x in P1^(S1\/S2);
    consider p, q be FinSequence such that
A2:   x = p^q & p in P1 & q in S1\/S2 by A1,POLNOT_1:def 2;
    q in S1 or q in S2 by A2,XBOOLE_0:def 3;
    then x in P1^S1 or x in P1^S2 by A2,POLNOT_1:def 2;
    hence thesis by XBOOLE_0:def 3;
  end;
A3: P1^S1 c= P1^(S1\/S2)
  proof
    let x be object such that
A4:   x in P1^S1;
    consider p, q be FinSequence such that
A5:   x = p^q & p in P1 & q in S1 by A4,POLNOT_1:def 2;
    q in S1\/ S2 by A5,XBOOLE_0:def 3;
    hence thesis by A5,POLNOT_1:def 2;
  end;
  P1^S2 c= P1^(S1\/S2)
  proof
    let x be object such that
A6:   x in P1^S2;
    consider p, q be FinSequence such that
A7:   x = p^q & p in P1 & q in S2 by A6,POLNOT_1:def 2;
    q in S1\/ S2 by A7,XBOOLE_0:def 3;
    hence thesis by A7,POLNOT_1:def 2;
  end;
  hence thesis by A3,XBOOLE_1:8;
end;
