
theorem Th1:
  for A, B being non empty set, f being Function of A,B, C being
  Subset of A, v being Element of B holds f +* (C-->v) is Function of A,B
proof
  let A, B be non empty set;
  let f be Function of A,B;
  let C be Subset of A;
  let v be Element of B;
A1: dom f = A by FUNCT_2:def 1;
  rng f c= B & rng(C-->v) c= {v} by FUNCOP_1:13,RELAT_1:def 19;
  then
A2: rng f \/ rng(C-->v) c= B \/ {v} by XBOOLE_1:13;
  rng(f +* (C-->v)) c= rng f \/ rng(C-->v) by FUNCT_4:17;
  then rng(f +* (C-->v)) c= B \/ {v} by A2,XBOOLE_1:1;
  then
A3: rng(f +* (C-->v)) c= B by ZFMISC_1:40;
  dom(f +* (C-->v)) = (dom f) \/ dom(C-->v) by FUNCT_4:def 1
    .= [#]A \/ C by A1,FUNCOP_1:13
    .= A by SUBSET_1:11;
  hence thesis by A3,FUNCT_2:def 1,RELSET_1:4;
end;
