
theorem Th18:
  for X,Y being set, f being Function holds (X \/ Y)-indexing f =
  (X-indexing f) +* (Y-indexing f)
proof
  let X,Y be set, f be Function;
  set Z = X \/ Y;
A1: f|Y c= f by RELAT_1:59;
  f|X c= f by RELAT_1:59;
  then f|X tolerates f|Y by A1,PARTFUN1:52;
  then
A2: (f|X) \/ (f|Y) = (f|X)+*(f|Y) by FUNCT_4:30;
  dom (f|X) = dom f /\ X by RELAT_1:61;
  then
A3: dom (f|X) c= dom f by XBOOLE_1:17;
  dom (f|Y) = dom f /\ Y by RELAT_1:61;
  then
A4: dom (f|X) /\ dom id Y c= dom (f|Y) by A3,XBOOLE_1:27;
  thus Z-indexing f = ((id X)+*id Y)+*(f|Z) by FUNCT_4:22
    .= ((id X)+*id Y)+*((f|X)+*(f|Y)) by A2,RELAT_1:78
    .= (id X)+*((id Y)+*((f|X)+*(f|Y))) by FUNCT_4:14
    .= (id X)+*((id Y)+*(f|X)+*(f|Y)) by FUNCT_4:14
    .= (id X)+*((f|X)+*(id Y)+*(f|Y)) by A4,Th1
    .= (id X)+*((f|X)+*((id Y)+*(f|Y))) by FUNCT_4:14
    .= (X-indexing f)+*(Y-indexing f) by FUNCT_4:14;
end;
