reserve V, C for set;
reserve A, B, D for Element of Fin PFuncs (V, C);
reserve s for Element of PFuncs (V,C);

theorem Th22:
  for K, L, M being Element of Fin PFuncs (V, C) holds K^(L \/ M) = K^L \/ K^M
proof
  let K, L, M be Element of Fin PFuncs (V, C);
  now
    let a be set;
    assume
A1: a in K^(L \/ M);
    then consider b,c being set such that
A2: b in K and
A3: c in L \/ M and
A4: a = b \/ c by Th15;
    K^(L \/ M) c= PFuncs (V, C) by FINSUB_1:def 5;
    then reconsider a9 = a as Element of PFuncs (V,C) by A1;
    K c= PFuncs (V, C) & L \/ M c= PFuncs (V, C) by FINSUB_1:def 5;
    then reconsider b9 = b, c9 = c as Element of PFuncs (V,C) by A2,A3;
    b9 c= a9 & c9 c= a9 by A4,XBOOLE_1:7;
    then
A5: b9 tolerates c9 by PARTFUN1:57;
    c9 in L or c9 in M by A3,XBOOLE_0:def 3;
    then a in K^L or a in K^M by A2,A4,A5;
    hence a in K^L \/ K^M by XBOOLE_0:def 3;
  end;
  hence K^(L \/ M) c= K^L \/ K^M;
  K^L c= K^(L \/ M) & K^M c= K^(L \/ M) by Th18,XBOOLE_1:7;
  hence thesis by XBOOLE_1:8;
end;
